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Roots Help

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By — McGraw-Hill Professional
Updated on Sep 27, 2011

Introduction to Square Roots

The square root of a number is the nonnegative number whose square is the root. For example 3 is the square root of 9 because 3 2 = 9.

Examples

Roots Examples

It may seem that negative numbers could be square roots. It is true that (–3) 2 = 9. But Roots Examples is the symbol for the nonnegative number whose square is 9. Sometimes we say that 3 is the principal square root of 9. When we speak of an even root, we mean the nonnegative root. In general, Roots Examples if b n = a. There is no problem with odd roots being negative numbers:

Roots Examples

If n is even, b is assumed to be the nonnegative root. Also even roots of negative numbers do not exist in the real number system. In this book, it is assumed that even roots will be taken only of nonnegative numbers. For instance in Roots Examples , it is assumed that x is not negative.

Root properties are similar to exponent properties.

Properties of Roots

Property 1

6

Roots Examples

We can take the product then the root or take the individual roots then the product.

Examples

Roots Examples

Property 1 only applies to multiplication. There is no similar property for addition (nor subtraction). A common mistake is to “simplify” the sum of two squares. For example Roots Examples is incorrect. The following example should give you an idea of why these two expressions are not equal. If there were the property Roots Examples , then we would have

Roots Examples

This could only be true if 10 2 = 58.

Property 2

Roots Examples

We can take the quotient then the root or the individual roots then the quotient.

Example

Roots Examples

Property 3

Roots Examples (Remember that if n is even, then a must not be negative.)

We can take the root then the power or the power then take the root.

Property 4

Roots Examples

Property 4 can be thought of as a root-power cancellation law.

Example

Roots Example

Find practice problems and solutions at Roots Practice Problems — Set 1.

Simplifying Roots - Using Properties of Roots

These properties can be used to simplify roots in the same way canceling is used to simplify fractions. For instance you normally would not leave Roots Solutions without simplifying it as 5 any more than you would leave Roots Solutions without reducing it to 3. In Roots Solutions if m is at least as large as n , then Roots Solutions can be simplified using Property Roots Solutions and Property 4 Roots Solutions

Examples

Roots Examples

Find practice problems and solutions at Roots Practice Problems — Set 2.

Simplyfing Roots - Perfect Squares as Factors

Numbers like 18, 48, and 50 are not perfect squares but they do have perfect squares as factors. Using the same properties, Roots Solutions and Roots Solutions , we can simplify quantities like Roots Solutions .

Examples

Roots Examples

Find practice problems and solutions at Roots Practice Problems — Set 3.

Simplifying Roots - Eliminating Roots in the Denominator of Fractions

Roots of fractions or fractions with a root in the denominator are not simplified. To eliminate roots in denominators, use the fact that Roots Solutions and that any nonzero number over itself is one. We will begin with square roots. If the denominator is a square root, multiply the fraction by the denominator over itself. This will force the new denominator to be a perfect square.

Examples

Roots Examples

Find practice problems and solutions at Roots Practice Problems — Set 4.

Simplifying Roots - Cube Roots and Higher

In the case of a cube (or higher) root, multiplying the fraction by the denominator over itself usually does not work. To eliminate the n th root in the denominator, we need to write the denominator as the nth root of some quantity to the nth power. For example, to simplify Roots Solutions we need a 5 3 under the cube root sign. There is only one 5 under the cube root. We need a total of three 5s, so we need two more 5s. Multiply 5 by 5 2 to get 5 3 :

Roots Solutions

When the denominator is written as a power (often the power is 1) subtract this power from the root. The factor will have this number as a power.

Examples

Roots Examples

Roots Examples

Find practice problems and solutions at Roots Practice Problems — Set 5.

More practice problems for this concept can be found at: Exponents and Roots Practice Test.

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