RadF - Evaluating Radical Expressions Lesson

Math_Lesson_TopBanner.png Evaluating Radical Expressions

Radical Expressions

Square roots are the inverse of squares.

For example, LaTeX: \sqrt[]{9}=\sqrt[]{3^2}=3√9=√32=3, and therefore the square root "operation" and square "operation" (a number raised to the 2nd power) cancel each other out. As you learned in the "Quadratics Revisited" unit, if you are taking the square root of a negative number, you get the imaginary number (i).

Cube roots are the inverse of cubes. Fhttp://cms.gavirtualschool.org/DEV15/Math/CCGPSAlgebra2/Softchalks/03_RationalandRadicalRelationships/ada-reference.gifor example, LaTeX: \sqrt[3]{8}=\sqrt[3]{2^3}=23√8=3√23=2 , and therefore the cube root"operation" and cube "operation" (a number raised to the 3rd power) cancel each other out. You can take the cube root of a negative number and your answer will be negative. NEVER use i with cube roots, or any other roots except square roots.

Parts of a Radical Expression

The nth root of a real number, x, can be written as the radical expression LaTeX: \sqrt[n]{x}n√x. See the following example to see the index, radical sign, and radicand. If there is not a number written for the "index", then the index is assumed to equal 2.

image of the equation x to the square root of n index: n radical: square root x: radicand

Real nth roots of x. Let n be an integer (n > 1) and let x be a real number.

When n is an even integer, the following are true:

  • x < 0: There is ZERO real nth roots
  • x = 0: There is ONE real nth roots: LaTeX: \sqrt[n]{0}=0n√0=0
  • x > 0: There are TWO real nth roots: LaTeX: \pm\sqrt[n]{x}=\pm x^{\frac{1}{n}}±n√x=±x1n

When n is an odd integer, the following are true:

  • x < 0: There is ONE real nth root: LaTeX: \pm\sqrt[n]{x}=x^{\frac{1}{n}}±n√x=x1n
  • x > 0: There is ONE real nth root: LaTeX: \pm\sqrt[n]{x}=x^{\frac{1}{n}}±n√x=x1n

Example: Find the nth roots

Find the indicated real nth root(s) of x.

n = 3 and x = -27

Because n = 3 is odd and x = -27 < 0, -27 has one real cube root. Because LaTeX: \left(-3\right)^3=-27(−3)3=−27, you can write = LaTeX: \sqrt[3]{-27}=-3\:or\:\left(-27\right)^{\frac{1}{3}}=-33√−27=−3or(−27)13=−3

n = 4 and x = 16

Because n = 4 is even and x = 16 > 0, 16 has two fourth roots. Because LaTeX: 2^4=1624=16, you can write LaTeX: \pm\sqrt[4]{16}=\pm2±4√16=±2

Sometimes a "basic" radical expression is not given; sometimes the power on the base number is something other than the number "1."

Therefore our "picture" of a radical expression is a little more detailed. Pay careful attention to the following diagram: LaTeX: \sqrt[n]{a^m}n√am.

The nth root of a real number, a, raised to the nth can also be written as the radical expression LaTeX: \left(\sqrt[n]{a}\right)^m(n√a)m. In this example , the index is n, the radical sign is the same, the radicand is a, and the power is m. If there is not a number written for the "index", then the index is assumed to equal 2. If there is not a number written for the "power," then the power is assumed to be equal to 1. There are times when it is beneficial to have the radical expression written in radical form, and there are times when it is more beneficial to have the radical expressions written in exponential form (when the expression is written with rational exponents ).

Example: Changing Between Exponential Form and Radical Form

  1. Re-write the expressions in radical form.

LaTeX: \left(8\right)^{\frac{3}{2}}=\left(\sqrt[2]{8}\right)^3=\left(\sqrt[]{8}\right)^3(8)32=(2√8)3=(√8)3

LaTeX: \left(43\right)^{\frac{3}{5}}=\left(\sqrt[5]{43}\right)^5(43)35=(5√43)5

LaTeX: \left(77\right)^{-\frac{3}{5}}-\frac{1}{\left(77\right)^{\frac{3}{5}}}=\frac{1}{\left(\sqrt[5]{77}\right)^3}(77)−35−1(77)35=1(5√77)3 (We will not consider rationalizing the denominator yet.)

2. Re-write the expression in exponential form.

LaTeX: \left(\sqrt[3]{13}\right)^2=\left(13\right)^{\frac{2}{3}}(3√13)2=(13)23

Example: Evaluating Expressions with Rational Exponents

Evaluate LaTeX: \left(9\right)^{\frac{3}{2}}\:and\:\left(243\right)^{-\frac{3}{5}}(9)32and(243)−35.

Evaluate Using Rational Exponent Form

LaTeX: \left(9\right)^{\frac{3}{2}}=\left(9^{\frac{1}{2}}\right)^3=3^3=27 \\ \left(243\right)^{-\frac{3}{5}}=\left(243^{\frac{1}{5}}\right)^{-3}=\frac{1}{\left(243^{\frac{1}{5}}\right)^3}=\frac{1}{3^3}=\frac{1}{27}(9)32=(912)3=33=27(243)−35=(24315)−3=1(24315)3=133=127

Evaluate Using Radical Form

LaTeX: \left(9\right)^{\frac{3}{2}}=\left(\sqrt[]{9}\right)^3=3^3=27 \\ \left(243\right)^{-\frac{3}{5}}=\left(\sqrt[5]{243}\right)^{-3}=\frac{1}{\left(\sqrt[5]{243}\right)^3}=\frac{1}{3^3}=\frac{1}{27}(9)32=(√9)3=33=27(243)−35=(5√243)−3=1(5√243)3=133=127

Example: Approximate Roots with a Calculator

Evaluate LaTeX: 8^{\frac{3}{2}},\:\left(43\right)^{-\frac{3}{5}},\:and\:\left(\sqrt[5]{13}\right)^3832,(43)−35,and(5√13)3

image of a chart depicting calculator strokes Expression: (a)(8) to the (3/2) Calc Stroke: 8^(3÷2) ENTER Display: 22.62741699797 Expression: (b)(43) to the (-3/5) Calc Stroke: 43^(3÷5) ENTER Display: 0.10469330901 Expression: (c)(fifth root of 13) to the (3) Calc Stroke: 13^(3÷5) ENTER Display: 4.659786420036

nth Roots, Exponential Form, Radical Form, and Rational Exponents Practice

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