Powers and Roots

Powers

Powers or exponents refer to multiplying the same number to itself a certain number of times. The same is true for variables and algebraic expressions that have exponents. The exponent will be located in the upper right hand corner next to the number and will be much smaller than the number (called superscript).

Positive and Negative Powers

When the power is represented as a positive integer, you simply multiply the number times itself the number of times designated by the power.

When the power is represented as a negative integer, you take the reciprocal of the number and multiply the reciprocal times itself the number of times designated by the power.

The negative in powers does not mean that the product will become negative; the negative only means you take the reciprocal of the integer. You may also take the number to its power first and then find the reciprocal of that result.

Powers in Fractions and Decimals

Any fraction or decimal taken to a power that is a positive integer will always equal a smaller number.

Any fraction or decimal taken to a power that is a negative integer will always equal a larger number.

Any number taken to the power of one will equal the original number. Any number raised to the power of zero will equal 1.

Multiplying and Dividing Powers with the Same Base

When multiplying numbers with the same base, but different exponents (negative or positive), such as 9² with 9³, you add the exponents together.

When dividing numbers of the same base with exponents (negative or positive), you subtract the exponent of the denominator from the exponent of the numerator.

Remember, when you divide another number by a fraction, you may multiply the number by the reciprocal of the fraction to achieve the correct answer.

When you take a number with a power to another power (negative or positive), you multiply the two powers together.

Roots

Roots take the opposite action of powers, in that the root of a number is another number multiplied by itself a certain number of times to make the original number, such as how 8 is the square root of 64 and 4 is the cube root of 64. 

The root can be written as the symbol √ (called a radical) and will encompass the original number.

You can find the root of a number through factoring. 

If you are looking for the square root of a number, you look for the same number multiplied to itself twice with no remaining numbers in the factor tree. 

If you are looking for the third (cube) root of a number, you look for the same number multiplied by itself three times with no remaining numbers in the factor tree (and so on).

Powers and roots may be represented together in a single fraction in the exponent, where the numerator is the power and the denominator is the root:

Multiplying and Dividing Roots

When multiplying numbers of the same base with fraction exponents, you add the fraction exponents as you would normal fractions.

When dividing numbers of the same base with fraction exponents, you subtract the fraction exponents as you would normal fractions.

Keep in Mind

A negative number taken to a power that is an even integer will result in a positive number; therefore, even roots will always have a positive and a negative solution:

A negative number taken to a power that is an odd integer will result in a negative number. For example,

The even root of a negative number is an imaginary number.

Table of Squares and Cubes Up to 20

Practice Problems

1. What is the value of 6³?
2. Simplify the expression .
3. What is the value of ?
4. Simplify the expression .
5. What is the value of ?

Practice Problem Answers

1. 216

2. x²y³

3. 12i

4. 9

5. 1/125