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Power and Root

You might want to recall the list of perfect square and cube roots while working through this lesson.

Perfect square roots:

  • the square root of 1
  • the square root of 4
  • the square root of 9
  • the square root of 16
  • the square root of 25
  • the square root of 36
  • the square root of 49
  • the square root of 64
  • the square root of 81
  • the square root of 100

Perfect cubed roots:

  • the cube root of 1 equals 1
  • the cube root of 8 equals 2
  • the cube root of 27 equals 3
  • the cube root of 64 equals 4
  • the cube root of 125 equals 5
  • the cube root of 216 equals 6
  • the cube root of 343 equals 7
  • the cube root of 512 equals 8
  • the cube root of 729 equals 9
  • the cube root of 1000 equals 10

Rational Exponents

A rational exponent can be rewritten as a radical. The numerator of the rational exponent will be the power of the radicand The radicand is the number under the radical. and the denominator will be the index The index is the type of root: square root or cube root .

You can also think of the rational exponent as power divided by the root.

x raised to the power over root power

Example #1

Identify the power and root for the rational exponent.

x raised to the power over root power

x raised to the six halves power

The power will be 6 and the root will be 2.

This can be rewritten as x raised to the six halves power equals the second root of x to the sixth power.

Power = 6, root = 2

Example #2

Identify the power and root for the rational exponent.

6 raised to the three halves power

The power of the rational exponent is 3 and the root is 2.

Rewrite using radical notation.

the second root of 6 cubed

Simplify 63 under the radical.

the square root of 6 cubed equals the square root of 216

Simplify the radical by rewriting 216 as the product of a perfect square and the remainder.

the square root of 6 cubed equals the square root of 216 = the square root of blank times 6

the square root of 6 cubed equals the square root of 216 equals the square root of 36 times 6 equals blank times the square root of 6

=

Example #3

Identify the power and root for the rational exponent.

27 raised to the one third power

Begin by writing the expression using radical notation.

What is the power?

What is the root?

27 raised to the one third power = the cube root of 27 to the first power

Simplify the radical.

27 raised to the one third power = the cube root of 27 =

Example #4

Watch Rewrite Radicals in Exponential Form.

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Example #5

Rewrite the radical using exponential notation.

the sixth root of 7 to the fourth power

What is the power?

What is the root?

Using the notation x raised to the power over root power, complete the expression.

x raised to the power over root power = 7 raised to the four sixths power

Simplify the rational exponent.

x raised to the power over root power = 7 raised to the four sixths power = 7 raised to the two thirds power

Example #6

Watch Simplify Rational Exponents by Rewriting in Radical Form.

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Rules of Exponents

Reviewing the Exponent Rules

Earlier in this course, you learned how to apply the properties of exponents. Review each of the following:

Product Rule of Exponents

35 x 32 = 35 + 2 = 37 = 2187

Power of a Power Rule of Exponents

(35)2 = 35(2) = 310 = 59049

Quotient Rule of Exponents

the fraction with numerator 3 to the fifth power and denominator 3 squared= 35 - 2 = 33 = 27

Example #7

Watch Apply Properties of Exponents to Simplify Expressions.

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Methods

When you are simplifying rational exponents, you can rewrite in radical notation to simplify or you can apply the properties of exponents to simplify.

Example #8

Watch Apply Properties of Exponents to Simplify Expressions.

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