Roots as rational or fractional exponents

• Introduction

• Roots as Powers, Properties of Powers, Important Property, Product and Quotient of Roots

• Solved Exercises: simplifying expressions with roots

Introduction

Power is an expression of this type

ab = a · a · · · a · a

This expression represents the result of multiplying the base, a, by itself as many times as the exponent, b, indicates. We read it as "a to the power of b".

In this page we are going to see cases where the exponent, b, is a fraction. In other words, we are going to work with roots and their powers.

Roots as Powers

Let n be a natural number different from zero (1, 2, 3, 4,...),

We will call the root of degree n or nth root of the number a to

$$\sqrt[n]{a} = a^\frac{1}{n} := b,\ b^n = a$$

In other words, the nth root of the number a is the number b, that to the power of n is a (so, b n = a).

The number n is called the degree of the root and a is called the radicand of the root.

Let's see some special cases:

• The root of degree n = 2 is known as a square root.

Example:

The square root of 9 is 3 because 3 to the power of two is 9.

$$\sqrt{9} = 3$$

• The root of degree n = 3 is known as a cube root.

Example:

The cube root of -8 is -2 because -2 to the power of three is -8.

$$\sqrt[3]{-8} = -2$$

Important: There are no roots with an even number degree (2, 4, 6, 8..) of negative numbers (they are complex numbers), but there are roots of negative numbers if the degree is an uneven number.

PROPERTIES OF POWERS

Product

Power

Quotient

Negative exponent

Inverse

Inverse of inverse

Important Property

The following property will probably be the one we will use the most:

Product and Quotient of Roots

The product of two roots with the same degree is the root (of same degree) of the product of the radicands, this is,

$$\sqrt[n]{a}\cdot \sqrt[n]{b} = \sqrt[n]{a\cdot b}$$

The same happens with the quotient:

Exercise 1

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Exercise 2

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Exercise 3

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Exercise 4

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Exercise 5

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Exercise 6

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Exercise 7

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Exercise 8

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Exercise 9

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Exercise 10

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Exercise 11

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Exercise 12

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Exercise 13

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Exercise 14

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Exercise 15

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