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.
It may seem that negative numbers could be square roots. It is true that (–3) 2 = 9. But 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, if b n = a. There is no problem with odd roots being negative numbers:
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 , it is assumed that x is not negative.
Root properties are similar to exponent properties.
Properties of Roots
We can take the product then the root or take the individual roots then the product.
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 is incorrect. The following example should give you an idea of why these two expressions are not equal. If there were the property , then we would have
This could only be true if 10 2 = 58.
We can take the quotient then the root or the individual roots then the quotient.
(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 can be thought of as a root-power cancellation law.
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 without simplifying it as 5 any more than you would leave without reducing it to 3. In if m is at least as large as n , then can be simplified using Property and Property 4
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, and , we can simplify quantities like .
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 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.
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 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 :
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.
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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