Powers and Exponents

Exponents

An exponent is a positive or negative number placed above and to the right of a quantity. It expresses the power to which the quantity is to be raised or lowered. In 4³, 3 is the exponent. It shows that 4 is to be used as a factor three times: 4 × 4 × 4 (multiplied by itself twice). 4³ is read as four to the third power (or four cubed).

Remember: x¹ = x and x⁰ = 1 when x is any number (other than 0).

Negative exponents

If the exponent is negative, such as 4⁻², the number and exponent may be dropped under the number 1 in a fraction to remove the negative sign.

Example 1: Simply the following by removing the exponents.

• (a)

  4⁻²

• (b)

  5⁻³

• (c)

  2⁻⁴

• (d)

  3⁻¹

Squares and cubes

Two specific types of powers should be noted: squares and cubes. To square a number, just multiply it by itself (the exponent is 2). For example, 6 squared (written 6²) is 6 × 6, or 36. 36 is called a perfect square (the square of a whole number). Following is a partial list of perfect squares:

To cube a number, just multiply it by itself twice (the exponent is 3). For example, 5 cubed (written 5³ is 5 × 5 × 5, or 125. 125 is called a perfect cube (the cube of a whole number). Following is a partial list of perfect cubes.

Operations with powers and exponents

To multiply two numbers with exponents, if the base numbers are the same, simply keep the base number and add the exponents.

Example 2: Multiply the following, leaving the answers with exponents.

• (a)

  2³ × 2⁵

• (b)

  3² × 3⁵

• (c)

  5⁴ × 5⁷

To divide two numbers with exponents, if the base numbers are the same, simply keep the base number and subtract the second exponent from the first, or the exponent of the denominator from the exponent of the numerator.

Example 3: Divide the following, leaving the answers with exponents.

To multiply or divide numbers with exponents, if the base numbers are different, you must simplify each number with an exponent first and then perform the operation.

Example 4: Simplify and perform the operation indicated.

• (a)

  2³ × 3²

• (b)

  6² ÷ 2³

  
  

  
  

For problems such as those in Example 4, some shortcuts are possible.

To add or subtract numbers with exponents, whether the base numbers are the same or different, you must simplify each number with an exponent first and then perform the indicated operation.

Example 5: Simplify and perform the operation indicated.

• (a)

  3² − 2³

• (b)

  5² + 3³

• (c)

  4² + 9³

• (d)

  2³ − 2²

• (a)

  3² − 2³ = 9 − 8 = 1

• (b)

  5² + 3³ = 25 + 27 = 52

• (c)

  4² + 9³ = 16 + 729 = 745

• (d)

  2³ − 2² = 8 − 4 = 4

If a number with an exponent is taken to another power (4²)³, simply keep the original base number and multiply the exponents.

Example 6: Multiply the following and leave the answers with exponents.

• (a)

  (6³)²

• (b)

  (3²)⁴

• (c)

  (5⁴)³

• (a)

  (6³)² = 6^(3×2) = 6⁶

• (b)

  (3²)⁴ = 3^(2×4) = 3⁸

• (c)

  (5⁴)³ = 5^(4×3) = 5¹²