Introduction to Radical Expressions:
Radical expressions involve roots, which are inverse operations of powers. For example, the square root of a number is the number that, when multiplied by itself, gives the original number. More generally, a radical expression involves a number or expression under a radical sign, commonly referred to as the “root” sign. The most common radical is the square root, but cube roots and higher-order roots also exist.
Radical expressions can be simplified, added, subtracted, multiplied, and divided. In this section, we will explore the rules and techniques for working with radicals and provide examples to illustrate each concept.
1. Definition of a Radical Expression:
A radical expression is any expression that contains a radical sign. The general form of a radical expression is:
$ \sqrt[n]{a} $
where $n$ is the degree of the root, and $a$ is the radicand (the expression under the radical).
If no index $n$ is specified, it is assumed to be 2 (i.e., a square root). For example:
$ \sqrt{25} = 5 $
because $5 \times 5 = 25$. Other examples include cube roots $ \sqrt[3]{x} $ and higher-order roots.
2. Properties of Radicals:
2.1 Multiplication of Radicals:
The product of two radicals is equivalent to the radical of the product:
$ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} $
Example 1:
Simplify $ \sqrt{4 \times 9} $.
$ \sqrt{4 \times 9} = \sqrt{4} \times \sqrt{9} $
$ = 2 \times 3 $
$ = 6 $
2.2 Division of Radicals:
The division of two radicals is equivalent to the radical of the quotient:
$ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} $
Example 2:
Simplify $ \frac{\sqrt{16}}{\sqrt{25}} $.
$ \frac{\sqrt{16}}{\sqrt{25}} = \sqrt{\frac{16}{25}} $
$ = \frac{4}{5} $
3. Simplifying Radical Expressions:
Radical expressions can often be simplified by factoring the radicand and applying the properties of radicals.
3.1 Simplifying Square Roots:
To simplify a square root, look for perfect square factors in the radicand.
$ \sqrt{a} = \sqrt{b^2 \times c} = b \times \sqrt{c} $
Example 3:
Simplify $ \sqrt{50} $.
$ \sqrt{50} = \sqrt{25 \times 2} $
$ = \sqrt{25} \times \sqrt{2} $
$ = 5 \times \sqrt{2} $
3.2 Simplifying Cube Roots:
Cube roots can also be simplified by finding perfect cube factors.
$ \sqrt[3]{a} = \sqrt[3]{b^3 \times c} = b \times \sqrt[3]{c} $
Example 4:
Simplify $ \sqrt[3]{54} $.
$ \sqrt[3]{54} = \sqrt[3]{27 \times 2} $
$ = \sqrt[3]{27} \times \sqrt[3]{2} $
$ = 3 \times \sqrt[3]{2} $
4. Adding and Subtracting Radicals:
Radicals can only be added or subtracted if they have the same radicand. For example:
$ a\sqrt{b} + c\sqrt{b} = (a + c)\sqrt{b} $
Example 5:
Simplify $ 3\sqrt{5} + 2\sqrt{5} $.
$ 3\sqrt{5} + 2\sqrt{5} = (3 + 2)\sqrt{5} $
$ = 5\sqrt{5} $
Example 6:
Simplify $ 4\sqrt{3} – \sqrt{3} $.
$ 4\sqrt{3} – \sqrt{3} = (4 – 1)\sqrt{3} $
$ = 3\sqrt{3} $
5. Solving Radical Equations:
When solving radical equations, the goal is to isolate the radical expression and then square both sides of the equation to eliminate the radical. Be sure to check for extraneous solutions after solving.
5.1 Solving Square Root Equations:
Example 7:
Solve $ \sqrt{x + 3} = 5 $.
Step 1: Square both sides.
$ (\sqrt{x + 3})^2 = 5^2 $
$ x + 3 = 25 $
Step 2: Solve for $x$.
$ x = 25 – 3 $
$ x = 22 $
5.2 Solving Cube Root Equations:
Example 8:
Solve $ \sqrt[3]{x – 1} = 4 $.
Step 1: Cube both sides.
$ (\sqrt[3]{x – 1})^3 = 4^3 $
$ x – 1 = 64 $
Step 2: Solve for $x$.
$ x = 64 + 1 $
$ x = 65 $
6. Rationalizing Denominators:
Rationalizing the denominator involves eliminating radicals from the denominator of a fraction. This is done by multiplying both the numerator and the denominator by a radical that will make the denominator a whole number.
6.1 Rationalizing Simple Denominators:
Example 9:
Simplify $ \frac{3}{\sqrt{2}} $.
Step 1: Multiply the numerator and denominator by $ \sqrt{2} $.
$ \frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2} $
6.2 Rationalizing Denominators with Binomials:
Example 10:
Simplify $ \frac{5}{\sqrt{3} + 2} $.
Step 1: Multiply both the numerator and denominator by the conjugate of the denominator.
$ \frac{5}{\sqrt{3} + 2} \times \frac{\sqrt{3} – 2}{\sqrt{3} – 2} = \frac{5(\sqrt{3} – 2)}{(\sqrt{3})^2 – (2)^2} $
Step 2: Simplify.
$ = \frac{5\sqrt{3} – 10}{3 – 4} $
$ = \frac{5\sqrt{3} – 10}{-1} $
$ = -5\sqrt{3} + 10 $
7. Multiplying Radicals:
Multiplying radicals follows the property:
$ \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} $
Example 11:
Simplify $ \sqrt{2} \times \sqrt{8} $.
$ \sqrt{2} \times \sqrt{8} = \sqrt{2 \times 8} $
$ = \sqrt{16} $
$ = 4 $
Example 12:
Simplify $ \sqrt{7} \times \sqrt{21} $.
$ \sqrt{7} \times \sqrt{21} = \sqrt{7 \times 21} $
$ = \sqrt{147} $
$ = \sqrt{49 \times 3} $
$ = 7\sqrt{3} $
8. Dividing Radicals:
Dividing radicals follows the property:
$ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} $
Example 13:
Simplify $ \frac{\sqrt{50}}{\sqrt{2}} $.
$ \frac{\sqrt{50}}{\sqrt{2}} = \sqrt{\frac{50}{2}} $
$ = \sqrt{25} $
$ = 5 $
Example 14:
Simplify $ \frac{\sqrt{75}}{\sqrt{3}} $.
$ \frac{\sqrt{75}}{\sqrt{3}} = \sqrt{\frac{75}{3}} $
$ = \sqrt{25} $
$ = 5 $
9. Combining Like Radicals:
Like terms with radicals are those that have the same radicand. You can only add or subtract like radicals.
$ a\sqrt{b} + c\sqrt{b} = (a + c)\sqrt{b} $
Example 15:
Simplify $ 2\sqrt{5} + 3\sqrt{5} $.
$ 2\sqrt{5} + 3\sqrt{5} = (2 + 3)\sqrt{5} $
$ = 5\sqrt{5} $
Example 16:
Simplify $ 4\sqrt{7} – 2\sqrt{7} $.
$ 4\sqrt{7} – 2\sqrt{7} = (4 – 2)\sqrt{7} $
$ = 2\sqrt{7} $
10. Solving Radical Equations:
10.1 Steps to Solve Radical Equations:
- Isolate the radical on one side of the equation.
- Square both sides to eliminate the radical.
- Solve the resulting equation.
- Check for extraneous solutions.
Example 17:
Solve $ \sqrt{x + 2} = 6 $.
Step 1: Square both sides.
$ (\sqrt{x + 2})^2 = 6^2 $
$ x + 2 = 36 $
Step 2: Solve for $x$.
$ x = 36 – 2 $
$ x = 34 $
Example 18:
Solve $ \sqrt{2x – 1} = 5 $.
Step 1: Square both sides.
$ (\sqrt{2x – 1})^2 = 5^2 $
$ 2x – 1 = 25 $
Step 2: Solve for $x$.
$ 2x = 25 + 1 $
$ 2x = 26 $
$ x = \frac{26}{2} $
$ x = 13 $
11. Radical Expressions Involving Higher Powers:
11.1 Higher Order Radicals:
In addition to square roots, we can also work with cube roots, fourth roots, etc. The properties for manipulating higher-order roots are similar to those for square roots.
Example 19:
Simplify $ \sqrt[3]{8x^6} $.
$ \sqrt[3]{8x^6} = \sqrt[3]{8} \times \sqrt[3]{x^6} $
$ = 2 \times x^2 $
$ = 2x^2 $
Example 20:
Simplify $ \sqrt[4]{16x^8} $.
$ \sqrt[4]{16x^8} = \sqrt[4]{16} \times \sqrt[4]{x^8} $
$ = 2 \times x^2 $
$ = 2x^2 $
12. Rationalizing Denominators with Higher Powers:
When dealing with higher powers, multiply the numerator and denominator by a radical that will result in a perfect power in the denominator.
Example 21:
Simplify $ \frac{1}{\sqrt[3]{4}} $.
Step 1: Multiply numerator and denominator by $ \sqrt[3]{16} $.
$ \frac{1}{\sqrt[3]{4}} \times \frac{\sqrt[3]{16}}{\sqrt[3]{16}} = \frac{\sqrt[3]{16}}{\sqrt[3]{64}} $
Step 2: Simplify.
$ = \frac{\sqrt[3]{16}}{4} $
13. Mixed Radical Expressions:
A mixed radical is an expression that combines an integer and a radical, for example, $ 3\sqrt{2} $ or $ 2\sqrt[3]{5} $. These radicals can also be simplified and manipulated in combination with other expressions.
13.1 Simplifying Mixed Radicals:
Example 22:
Simplify $ 5\sqrt{18} $.
Step 1: Factor the radicand.
$ 5\sqrt{18} = 5\sqrt{9 \times 2} $
$ = 5 \times 3\sqrt{2} $
$ = 15\sqrt{2} $
13.2 Operations with Mixed Radicals:
Example 23:
Simplify $ 4\sqrt{6} + 3\sqrt{6} $.
$ 4\sqrt{6} + 3\sqrt{6} = (4 + 3)\sqrt{6} $
$ = 7\sqrt{6} $
14. Solving Radical Inequalities:
Radical inequalities involve finding the values of the variable that make the inequality true. The steps to solve a radical inequality are similar to those used to solve a radical equation.
Example 24:
Solve $ \sqrt{x + 1} \leq 3 $.
Step 1: Square both sides.
$ (\sqrt{x + 1})^2 \leq 3^2 $
$ x + 1 \leq 9 $
Step 2: Solve for $x$.
$ x \leq 9 – 1 $
$ x \leq 8 $
Step 3: Check for domain restrictions.
Since $ \sqrt{x + 1} $ is defined for $ x \geq -1 $, the solution is $ -1 \leq x \leq 8 $.
Example 25:
Solve $ \sqrt{2x – 3} > 4 $.
Step 1: Square both sides.
$ (\sqrt{2x – 3})^2 > 4^2 $
$ 2x – 3 > 16 $
Step 2: Solve for $x$.
$ 2x > 16 + 3 $
$ 2x > 19 $
$ x > \frac{19}{2} $
$ x > 9.5 $
15. Applications of Radical Equations:
Radical equations often appear in real-life problems such as physics, engineering, and biology.
15.1 Application: Free Fall Equation
The distance $d$ that an object falls in $t$ seconds under the influence of gravity is given by the equation:
$ d = \frac{1}{2}gt^2 $
where $g$ is the acceleration due to gravity ($9.8 , \text{m/s}^2$ on Earth).
Example 26:
An object is dropped from a height of 50 meters. How long does it take to hit the ground?
We know $d = 50$ meters and $g = 9.8 , \text{m/s}^2$. Solving for $t$:
$ 50 = \frac{1}{2}(9.8)t^2 $
$ 50 = 4.9t^2 $
$ t^2 = \frac{50}{4.9} $
$ t^2 \approx 10.204 $
$ t \approx \sqrt{10.204} $
$ t \approx 3.19 $ seconds
16. More Applications of Radical Equations:
16.1 Application: Pythagorean Theorem
The Pythagorean Theorem relates the lengths of the sides of a right triangle. It is given by:
$ a^2 + b^2 = c^2 $
where $c$ is the length of the hypotenuse, and $a$ and $b$ are the lengths of the other two sides.
Example 27:
A right triangle has legs of length 6 and 8. Find the length of the hypotenuse.
$ 6^2 + 8^2 = c^2 $
$ 36 + 64 = c^2 $
$ c^2 = 100 $
$ c = \sqrt{100} $
$ c = 10 $
Example 28:
A right triangle has a hypotenuse of 13 and one leg of 5. Find the length of the other leg.
$ a^2 + 5^2 = 13^2 $
$ a^2 + 25 = 169 $
$ a^2 = 169 – 25 $
$ a^2 = 144 $
$ a = \sqrt{144} $
$ a = 12 $
16.2 Application: Distance Formula
The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:
$ d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} $
Example 29:
Find the distance between the points $(2, 3)$ and $(7, 1)$.
$ d = \sqrt{(7 – 2)^2 + (1 – 3)^2} $
$ = \sqrt{5^2 + (-2)^2} $
$ = \sqrt{25 + 4} $
$ = \sqrt{29} $
$ \approx 5.39 $
Example 30:
Find the distance between the points $(5, -4)$ and $(-2, 6)$.
$ d = \sqrt{(-2 – 5)^2 + (6 + 4)^2} $
$ = \sqrt{(-7)^2 + 10^2} $
$ = \sqrt{49 + 100} $
$ = \sqrt{149} $
$ \approx 12.21 $
17. Simplifying Radicals Containing Variables:
When radicals contain variables, the same rules for simplifying apply. However, remember to treat exponents carefully.
Example 31:
Simplify $ \sqrt{16x^4} $.
$ \sqrt{16x^4} = \sqrt{16} \times \sqrt{x^4} $
$ = 4x^2 $
Example 32:
Simplify $ \sqrt{25y^6} $.
$ \sqrt{25y^6} = \sqrt{25} \times \sqrt{y^6} $
$ = 5y^3 $
18. Combining Radicals with Variables:
You can only combine radicals if they have the same radicand and variable part.
Example 33:
Simplify $ 3\sqrt{2x} + 5\sqrt{2x} $.
$ 3\sqrt{2x} + 5\sqrt{2x} = (3 + 5)\sqrt{2x} $
$ = 8\sqrt{2x} $
Example 34:
Simplify $ 7\sqrt{3y^2} – 4\sqrt{3y^2} $.
$ 7\sqrt{3y^2} – 4\sqrt{3y^2} = (7 – 4)\sqrt{3y^2} $
$ = 3\sqrt{3y^2} $
19. Radical Equations Involving Variables:
When solving radical equations involving variables, isolate the radical and then square both sides.
Example 35:
Solve $ \sqrt{2x + 5} = 7 $.
Step 1: Square both sides.
$ (\sqrt{2x + 5})^2 = 7^2 $
$ 2x + 5 = 49 $
Step 2: Solve for $x$.
$ 2x = 49 – 5 $
$ 2x = 44 $
$ x = \frac{44}{2} $
$ x = 22 $
Example 36:
Solve $ \sqrt{3x + 4} = 8 $.
Step 1: Square both sides.
$ (\sqrt{3x + 4})^2 = 8^2 $
$ 3x + 4 = 64 $
Step 2: Solve for $x$.
$ 3x = 64 – 4 $
$ 3x = 60 $
$ x = \frac{60}{3} $
$ x = 20 $
20. Solving Equations with Radicals on Both Sides:
When there are radicals on both sides of the equation, square both sides to eliminate the radicals.
Example 37:
Solve $ \sqrt{x + 1} = \sqrt{2x – 3} $.
Step 1: Square both sides.
$ (\sqrt{x + 1})^2 = (\sqrt{2x – 3})^2 $
$ x + 1 = 2x – 3 $
Step 2: Solve for $x$.
$ x – 2x = -3 – 1 $
$ -x = -4 $
$ x = 4 $
Example 38:
Solve $ \sqrt{2x + 5} = \sqrt{3x – 1} $.
Step 1: Square both sides.
$ (\sqrt{2x + 5})^2 = (\sqrt{3x – 1})^2 $
$ 2x + 5 = 3x – 1 $
Step 2: Solve for $x$.
$ 2x – 3x = -1 – 5 $
$ -x = -6 $
$ x = 6 $
21. Operations with Cube Roots:
Cube roots behave similarly to square roots in terms of operations like multiplication and division.
Example 39:
Simplify $ \sqrt[3]{8x^9} $.
$ \sqrt[3]{8x^9} = \sqrt[3]{8} \times \sqrt[3]{x^9} $
$ = 2x^3 $
Example 40:
Simplify $ \sqrt[3]{64y^6} $.
$ \sqrt[3]{64y^6} = \sqrt[3]{64} \times \sqrt[3]{y^6} $
$ = 4y^2 $
22. Rationalizing the Denominator with Cube Roots:
Just like with square roots, you can rationalize the denominator with cube roots by multiplying both the numerator and denominator by a cube root that will simplify the denominator.
Example 41:
Simplify $ \frac{1}{\sqrt[3]{4}} $.
Step 1: Multiply numerator and denominator by $ \sqrt[3]{16} $.
$ \frac{1}{\sqrt[3]{4}} \times \frac{\sqrt[3]{16}}{\sqrt[3]{16}} = \frac{\sqrt[3]{16}}{\sqrt[3]{64}} $
Step 2: Simplify.
$ = \frac{\sqrt[3]{16}}{4} $
23. Radical Expressions in Scientific Notation:
When simplifying radical expressions involving numbers in scientific notation, handle the coefficients and the powers of 10 separately.
Example 42:
Simplify $ \sqrt{4 \times 10^6} $.
$ \sqrt{4 \times 10^6} = \sqrt{4} \times \sqrt{10^6} $
$ = 2 \times 10^3 $
$ = 2000 $
Example 43:
Simplify $ \sqrt{9 \times 10^8} $.
$ \sqrt{9 \times 10^8} = \sqrt{9} \times \sqrt{10^8} $
$ = 3 \times 10^4 $
$ = 30000 $
24. Simplifying Expressions with Higher Order Roots:
When simplifying expressions with higher-order roots, the same rules for square roots and cube roots apply, but with higher indices.
Example 44:
Simplify $ \sqrt[4]{16x^8} $.
$ \sqrt[4]{16x^8} = \sqrt[4]{16} \times \sqrt[4]{x^8} $
$ = 2x^2 $
Example 45:
Simplify $ \sqrt[5]{32y^{10}} $.
$ \sqrt[5]{32y^{10}} = \sqrt[5]{32} \times \sqrt[5]{y^{10}} $
$ = 2y^2 $
25. Combining Higher-Order Roots:
Just like with square roots and cube roots, you can combine higher-order roots if they have the same index and radicand.
Example 46:
Simplify $ 3\sqrt[4]{5x} + 2\sqrt[4]{5x} $.
$ 3\sqrt[4]{5x} + 2\sqrt[4]{5x} = (3 + 2)\sqrt[4]{5x} $
$ = 5\sqrt[4]{5x} $
Example 47:
Simplify $ 7\sqrt[3]{2y^2} – 4\sqrt[3]{2y^2} $.
$ 7\sqrt[3]{2y^2} – 4\sqrt[3]{2y^2} = (7 – 4)\sqrt[3]{2y^2} $
$ = 3\sqrt[3]{2y^2} $
26. Solving Higher-Order Radical Equations:
The process for solving higher-order radical equations is the same as for square and cube roots. Isolate the radical, and raise both sides to the appropriate power.
Example 48:
Solve $ \sqrt[4]{x + 5} = 3 $.
Step 1: Raise both sides to the 4th power.
$ (\sqrt[4]{x + 5})^4 = 3^4 $
$ x + 5 = 81 $
Step 2: Solve for $x$.
$ x = 81 – 5 $
$ x = 76 $
Example 49:
Solve $ \sqrt[5]{2x – 3} = 4 $.
Step 1: Raise both sides to the 5th power.
$ (\sqrt[5]{2x – 3})^5 = 4^5 $
$ 2x – 3 = 1024 $
Step 2: Solve for $x$.
$ 2x = 1024 + 3 $
$ 2x = 1027 $
$ x = \frac{1027}{2} $
$ x = 513.5 $
27. Radical Expressions in Word Problems:
Radical expressions often appear in real-world problems, such as those involving geometry or physics.
Example 50:
The formula for the area $A$ of an equilateral triangle with side length $s$ is given by:
$ A = \frac{\sqrt{3}}{4} s^2 $
Find the area of an equilateral triangle with a side length of 8.
$ A = \frac{\sqrt{3}}{4} (8^2) $
$ A = \frac{\sqrt{3}}{4} \times 64 $
$ A = 16\sqrt{3} $
Example 51:
The formula for the period $T$ of a pendulum with length $L$ is given by:
$ T = 2\pi\sqrt{\frac{L}{g}} $
where $g = 9.8 , m/s^2$. Find the period of a pendulum with length $L = 4$ meters.
$ T = 2\pi\sqrt{\frac{4}{9.8}} $
$ T = 2\pi\sqrt{0.4082} $
$ T \approx 2\pi \times 0.639 $
$ T \approx 4.016 $
28. Rationalizing Higher-Order Roots:
Rationalizing the denominator with higher-order roots works in a similar way to square roots. Multiply the numerator and denominator by a root that will simplify the denominator.
Example 52:
Simplify $ \frac{1}{\sqrt[4]{2}} $.
Step 1: Multiply numerator and denominator by $ \sqrt[4]{16} $.
$ \frac{1}{\sqrt[4]{2}} \times \frac{\sqrt[4]{16}}{\sqrt[4]{16}} = \frac{\sqrt[4]{16}}{\sqrt[4]{32}} $
Step 2: Simplify.
$ = \frac{\sqrt[4]{16}}{2} $
Example 53:
Simplify $ \frac{3}{\sqrt[5]{3}} $.
Step 1: Multiply numerator and denominator by $ \sqrt[5]{81} $.
$ \frac{3}{\sqrt[5]{3}} \times \frac{\sqrt[5]{81}}{\sqrt[5]{81}} = \frac{3\sqrt[5]{81}}{\sqrt[5]{243}} $
Step 2: Simplify.
$ = \frac{3\sqrt[5]{81}}{3} $
$ = \sqrt[5]{81} $
29. Operations with Radicals and Exponents:
Radicals and exponents can be combined using the rule that $ \sqrt[n]{a^m} = a^{\frac{m}{n}} $.
Example 54:
Simplify $ \sqrt[3]{x^6} $.
$ \sqrt[3]{x^6} = x^{\frac{6}{3}} $
$ = x^2 $
Example 55:
Simplify $ \sqrt[5]{y^{10}} $.
$ \sqrt[5]{y^{10}} = y^{\frac{10}{5}} $
$ = y^2 $
30. Multiplying and Dividing Radicals with Variables:
When multiplying and dividing radicals that contain variables, apply the same rules for numbers and then simplify.
Example 56:
Simplify $ \sqrt{x} \times \sqrt{x^3} $.
$ \sqrt{x} \times \sqrt{x^3} = \sqrt{x \times x^3} $
$ = \sqrt{x^4} $
$ = x^2 $
Example 57:
Simplify $ \frac{\sqrt{y^5}}{\sqrt{y}} $.
$ \frac{\sqrt{y^5}}{\sqrt{y}} = \sqrt{\frac{y^5}{y}} $
$ = \sqrt{y^4} $
$ = y^2 $
31. Adding and Subtracting Radicals with Variables:
You can only add or subtract radicals with the same radicand and variable part.
Example 58:
Simplify $ 4\sqrt{x} + 7\sqrt{x} $.
$ 4\sqrt{x} + 7\sqrt{x} = (4 + 7)\sqrt{x} $
$ = 11\sqrt{x} $
Example 59:
Simplify $ 8\sqrt{y^2} – 3\sqrt{y^2} $.
$ 8\sqrt{y^2} – 3\sqrt{y^2} = (8 – 3)\sqrt{y^2} $
$ = 5\sqrt{y^2} $
$ = 5y $
32. Rational Exponents and Radicals:
When a radical is expressed as a rational exponent, such as $ a^{\frac{m}{n}} $, it can be written as a radical expression $ \sqrt[n]{a^m} $.
Example 60:
Simplify $ 27^{\frac{2}{3}} $.
$ 27^{\frac{2}{3}} = \sqrt[3]{27^2} $
$ = \sqrt[3]{729} $
$ = 9 $
Example 61:
Simplify $ 16^{\frac{3}{4}} $.
$ 16^{\frac{3}{4}} = \sqrt[4]{16^3} $
$ = \sqrt[4]{4096} $
$ = 8 $
33. Radical Equations in Context:
Radical equations often appear in real-world situations, such as in physics and engineering problems.
Example 62:
The speed $v$ of a wave on a string is given by $ v = \sqrt{\frac{T}{\mu}} $, where $T$ is the tension in the string and $ \mu $ is the linear mass density. Find the speed of a wave if $ T = 50 , N $ and $ \mu = 0.2 , kg/m $.
$ v = \sqrt{\frac{50}{0.2}} $
$ v = \sqrt{250} $
$ v \approx 15.81 , m/s $
Example 63:
The period $T$ of a satellite orbiting Earth is given by $ T = 2\pi\sqrt{\frac{r^3}{GM}} $, where $r$ is the distance from the center of the Earth, $G$ is the gravitational constant, and $M$ is the mass of the Earth. Find the period of a satellite that is $42,000$ km from the center of the Earth, assuming $ G = 6.67 \times 10^{-11} , Nm^2/kg^2 $ and $ M = 5.97 \times 10^{24} , kg $.
$ T = 2\pi\sqrt{\frac{(42,000 \times 10^3)^3}{6.67 \times 10^{-11} \times 5.97 \times 10^{24}}} $
$ \approx 86,164 $ seconds
$ \approx 23.9 $ hours
34. Further Examples of Simplifying Higher Order Radicals:
Example 64:
Simplify $ \sqrt[6]{64x^{12}} $.
$ \sqrt[6]{64x^{12}} = \sqrt[6]{64} \times \sqrt[6]{x^{12}} $
$ = 2x^2 $
Example 65:
Simplify $ \sqrt[4]{81y^{8}} $.
$ \sqrt[4]{81y^{8}} = \sqrt[4]{81} \times \sqrt[4]{y^{8}} $
$ = 3y^2 $
35. Simplifying Radicals with Fractions:
When simplifying radicals with fractions, treat the numerator and denominator separately, simplifying them under the same radical.
Example 66:
Simplify $ \sqrt{\frac{49}{64}} $.
$ \sqrt{\frac{49}{64}} = \frac{\sqrt{49}}{\sqrt{64}} $
$ = \frac{7}{8} $
Example 67:
Simplify $ \sqrt{\frac{36x^4}{49y^6}} $.
$ \sqrt{\frac{36x^4}{49y^6}} = \frac{\sqrt{36x^4}}{\sqrt{49y^6}} $
$ = \frac{6x^2}{7y^3} $
36. Rationalizing Denominators with Variables:
Rationalizing the denominator means eliminating any radicals in the denominator by multiplying both the numerator and denominator by an appropriate radical expression.
Example 68:
Rationalize the denominator of $ \frac{1}{\sqrt{2x}} $.
$ \frac{1}{\sqrt{2x}} \times \frac{\sqrt{2x}}{\sqrt{2x}} = \frac{\sqrt{2x}}{2x} $
Example 69:
Rationalize the denominator of $ \frac{3}{\sqrt{y} + 2} $.
$ \frac{3}{\sqrt{y} + 2} \times \frac{\sqrt{y} – 2}{\sqrt{y} – 2} = \frac{3(\sqrt{y} – 2)}{(\sqrt{y})^2 – (2)^2} $
$ = \frac{3(\sqrt{y} – 2)}{y – 4} $
37. Operations on Radicals with Complex Numbers:
When dealing with radicals and complex numbers, you apply the same principles but treat the imaginary unit $i = \sqrt{-1}$ separately.
Example 70:
Simplify $ \sqrt{-25} $.
$ \sqrt{-25} = 5i $
Example 71:
Simplify $ \sqrt{-64} $.
$ \sqrt{-64} = 8i $
38. Multiplying Radicals with Complex Numbers:
You can multiply radicals with complex numbers by multiplying the numbers and the $i$ terms separately.
Example 72:
Simplify $ \sqrt{-4} \times \sqrt{-9} $.
$ \sqrt{-4} \times \sqrt{-9} = 2i \times 3i $
$ = 6i^2 = 6(-1) = -6 $
Example 73:
Simplify $ \sqrt{-16} \times \sqrt{-25} $.
$ \sqrt{-16} \times \sqrt{-25} = 4i \times 5i $
$ = 20i^2 = 20(-1) = -20 $
39. Division of Radicals with Complex Numbers:
When dividing radicals with complex numbers, divide the numbers and then simplify the imaginary units separately.
Example 74:
Simplify $ \frac{\sqrt{-9}}{\sqrt{-4}} $.
$ \frac{\sqrt{-9}}{\sqrt{-4}} = \frac{3i}{2i} = \frac{3}{2} $
Example 75:
Simplify $ \frac{\sqrt{-25}}{\sqrt{-16}} $.
$ \frac{\sqrt{-25}}{\sqrt{-16}} = \frac{5i}{4i} = \frac{5}{4} $
40. Solving Radical Equations with Complex Numbers:
Solving radical equations that involve complex numbers follows the same process as solving real-number radical equations.
Example 76:
Solve $ \sqrt{x + 9} = \sqrt{-16} $.
Step 1: Simplify $ \sqrt{-16} = 4i $.
Step 2: Square both sides:
$ (\sqrt{x + 9})^2 = (4i)^2 $
$ x + 9 = 16(-1) $
$ x + 9 = -16 $
Step 3: Solve for $x$:
$ x = -16 – 9 = -25 $
Example 77:
Solve $ \sqrt{x + 6} = \sqrt{-36} $.
Step 1: Simplify $ \sqrt{-36} = 6i $.
Step 2: Square both sides:
$ (\sqrt{x + 6})^2 = (6i)^2 $
$ x + 6 = 36(-1) $
$ x + 6 = -36 $
Step 3: Solve for $x$:
$ x = -36 – 6 = -42 $
41. Simplifying Radicals in Scientific Notation:
Radical expressions can also involve numbers in scientific notation. When simplifying, treat the base numbers and the powers of 10 separately.
Example 78:
Simplify $ \sqrt{9 \times 10^4} $.
$ \sqrt{9 \times 10^4} = \sqrt{9} \times \sqrt{10^4} $
$ = 3 \times 10^2 $
$ = 300 $
Example 79:
Simplify $ \sqrt{4 \times 10^6} $.
$ \sqrt{4 \times 10^6} = \sqrt{4} \times \sqrt{10^6} $
$ = 2 \times 10^3 $
$ = 2000 $
42. Using Radicals in Geometry:
Radicals are often used in geometry to find exact values for lengths and areas.
Example 80:
Find the length of the diagonal of a square with a side length of 5.
The diagonal $d$ of a square is given by:
$ d = \sqrt{2} \times \text{side length} $
$ d = \sqrt{2} \times 5 $
$ d = 5\sqrt{2} $
Example 81:
Find the area of an equilateral triangle with side length 7.
The area $A$ of an equilateral triangle is given by:
$ A = \frac{\sqrt{3}}{4} \times \text{side length}^2 $
$ A = \frac{\sqrt{3}}{4} \times 7^2 $
$ A = \frac{\sqrt{3}}{4} \times 49 $
$ A = \frac{49\sqrt{3}}{4} $
43. Using Radicals in Physics:
Radical expressions also appear frequently in physics, particularly in formulas for energy and motion.
Example 82:
The formula for kinetic energy $KE$ is given by:
$ KE = \frac{1}{2}mv^2 $
Solve for $v$ in terms of $KE$ and $m$.
$ v = \sqrt{\frac{2KE}{m}} $
Example 83:
The period $T$ of a pendulum is given by:
$ T = 2\pi\sqrt{\frac{L}{g}} $
where $L$ is the length of the pendulum and $g$ is the acceleration due to gravity. Find $L$ in terms of $T$ and $g$.
$ L = \frac{T^2g}{4\pi^2} $
44. Estimating Radicals:
When exact values are not possible, radicals can be estimated to decimal form.
Example 84:
Estimate $ \sqrt{50} $.
Using a calculator:
$ \sqrt{50} \approx 7.0711 $
Example 85:
Estimate $ \sqrt[3]{200} $.
Using a calculator:
$ \sqrt[3]{200} \approx 5.8480 $
45. Using Radicals in Pythagorean Theorem:
Radicals often appear when using the Pythagorean theorem to find the lengths of sides in right triangles.
Example 86:
Find the length of the hypotenuse of a right triangle with legs of 6 and 8.
Using the Pythagorean theorem:
$ c = \sqrt{a^2 + b^2} $
$ c = \sqrt{6^2 + 8^2} $
$ c = \sqrt{36 + 64} $
$ c = \sqrt{100} $
$ c = 10 $
Example 87:
Find the length of one leg of a right triangle if the hypotenuse is 13 and the other leg is 12.
$ a = \sqrt{c^2 – b^2} $
$ a = \sqrt{13^2 – 12^2} $
$ a = \sqrt{169 – 144} $
$ a = \sqrt{25} $
$ a = 5 $
46. Simplifying Radicals with Fractions and Exponents:
Combining both fractions and exponents with radicals can require multiple steps to simplify.
Example 88:
Simplify $ \sqrt{\frac{x^4}{16}} $.
$ \sqrt{\frac{x^4}{16}} = \frac{\sqrt{x^4}}{\sqrt{16}} $
$ = \frac{x^2}{4} $
Example 89:
Simplify $ \sqrt[3]{\frac{8y^6}{27}} $.
$ \sqrt[3]{\frac{8y^6}{27}} = \frac{\sqrt[3]{8}}{\sqrt[3]{27}} \times \sqrt[3]{y^6} $
$ = \frac{2}{3} \times y^2 $
$ = \frac{2y^2}{3} $
47. Solving Radical Equations:
Radical equations are solved by isolating the radical and then squaring both sides of the equation.
Example 90:
Solve $ \sqrt{x + 4} = 6 $.
Step 1: Square both sides:
$ (\sqrt{x + 4})^2 = 6^2 $
$ x + 4 = 36 $
Step 2: Solve for $x$:
$ x = 36 – 4 = 32 $
Example 91:
Solve $ \sqrt{2x – 5} = 7 $.
Step 1: Square both sides:
$ (\sqrt{2x – 5})^2 = 7^2 $
$ 2x – 5 = 49 $
Step 2: Solve for $x$:
$ 2x = 49 + 5 = 54 $
$ x = \frac{54}{2} = 27 $
48. Verifying Solutions to Radical Equations:
It’s important to verify the solutions to radical equations because squaring both sides can sometimes introduce extraneous solutions.
Example 92:
Solve and verify $ \sqrt{x – 2} = x – 4 $.
Step 1: Square both sides:
$ (\sqrt{x – 2})^2 = (x – 4)^2 $
$ x – 2 = x^2 – 8x + 16 $
Step 2: Rearrange the equation:
$ 0 = x^2 – 9x + 18 $
Step 3: Solve the quadratic:
$ x = \frac{9 \pm \sqrt{81 – 72}}{2} $
$ x = \frac{9 \pm \sqrt{9}}{2} $
$ x = \frac{9 \pm 3}{2} $
$ x = 6 $ or $ x = 3 $
Step 4: Verify both solutions:
For $ x = 6 $:
$ \sqrt{6 – 2} = 6 – 4 $
$ 2 = 2 $, valid solution.
For $ x = 3 $:
$ \sqrt{3 – 2} = 3 – 4 $
$ 1 \neq -1 $, extraneous solution.
Final solution: $ x = 6 $
Example 93:
Solve and verify $ \sqrt{x + 1} = x – 2 $.
Step 1: Square both sides:
$ (\sqrt{x + 1})^2 = (x – 2)^2 $
$ x + 1 = x^2 – 4x + 4 $
Step 2: Rearrange the equation:
$ 0 = x^2 – 5x + 3 $
Step 3: Solve the quadratic:
$ x = \frac{5 \pm \sqrt{25 – 12}}{2} $
$ x = \frac{5 \pm \sqrt{13}}{2} $
Since $ \sqrt{13} $ is an irrational number, the exact solutions are $ x = \frac{5 + \sqrt{13}}{2} $ and $ x = \frac{5 – \sqrt{13}}{2} $.
Verifying these solutions can be done numerically or symbolically, and we only accept solutions that fit both sides of the equation.
49. Applying Radicals in Real-World Scenarios:
Radicals are used in various real-world applications, from physics to economics. Here’s one more example from finance.
Example 94:
The formula for compound interest is:
$ A = P(1 + \frac{r}{n})^{nt} $
Where $P$ is the principal amount, $r$ is the annual interest rate, $n$ is the number of times the interest is compounded per year, and $t$ is the time in years. Rearrange to solve for $P$ given $A$, $r$, $n$, and $t$.
$ P = \frac{A}{(1 + \frac{r}{n})^{nt}} $
Example 95:
In kinematics, the displacement $s$ of an object is given by:
$ s = ut + \frac{1}{2}at^2 $
where $u$ is the initial velocity, $a$ is the acceleration, and $t$ is the time. Rearrange to solve for $t$.
$ t = \frac{-u \pm \sqrt{u^2 + 2as}}{a} $
50. Final Examples of Simplifying Higher Order Radicals:
Example 96:
Simplify $ \sqrt[3]{8x^9} $.
$ \sqrt[3]{8x^9} = \sqrt[3]{8} \times \sqrt[3]{x^9} $
$ = 2x^3 $
Example 97:
Simplify $ \sqrt[4]{16y^{12}} $.
$ \sqrt[4]{16y^{12}} = \sqrt[4]{16} \times \sqrt[4]{y^{12}} $
$ = 2y^3 $
Example 98:
Simplify $ \sqrt[5]{32z^{10}} $.
$ \sqrt[5]{32z^{10}} = \sqrt[5]{32} \times \sqrt[5]{z^{10}} $
$ = 2z^2 $
Example 99:
Simplify $ \sqrt[3]{125x^{6}} $.
$ \sqrt[3]{125x^{6}} = \sqrt[3]{125} \times \sqrt[3]{x^6} $
$ = 5x^2 $
Example 100:
Simplify $ \sqrt[4]{81y^{16}} $.
$ \sqrt[4]{81y^{16}} = \sqrt[4]{81} \times \sqrt[4]{y^{16}} $
$ = 3y^4 $
