Skill Builder: Exponent Rules

Exponents may seem rather straightforward and simple. And that's usually the case, particularly when dealing with common exponents like 2 and 3.

Example: 3² = 3 x 3 = 9

There's no confusion here. To square a number simply means to multiply the number by itself.

Likewise, to cube a number follows the same logic and simply means to multiply the number by itself three times.

Example: 3³ = 3 x 3 x 3 = 27

But what about exponents that are 0? Negative? Or fractions? What do those exponents mean?

The following question from the 2020 SHSAT Practice Test B is great review of how exponents work.

Question 103. Given: 0³ + 4⁰ + 2⁻¹ + (−1)²

What is the value of the expression above, in decimal form?

A) 0

B) 0.5

C) 1.5

D) 2.5

A number raised to the 0 power

Example: 4⁰ ≠ 0

A number raised to the 0 power isn't equal to 0. Raising a number or a variable to the 0 power is always equal to 1.

Therefore, 4⁰ = 1 and x⁰ = 1

That said, 0 raised to any power, as is the first term in the above question, is equal to 0.

Example : 0³ = 0 x 0 x 0 = 0

Negative Exponents

A negative exponent has a different meaning. It doesn't mean the number changes to a negative value. A negative exponent means take the reciprocal (flip the fraction), and then make the exponent positive.

Example 1

2⁻³ = (½)³ = 1/8

Example 2

5⁻² = (⅕)² = 1/25

Example 3

(¾)⁻² = (4/3)² = 16/9

Recap

Let's return to the original SHSAT practice question and put our exponent review to work.

Given: 0³ + 4⁰ + 2⁻¹ + (−1)²

0³ = 0 x 0 x0 = 0

4⁰ = 1

2⁻¹ = (½)¹ = ½

−1² = -1 x -1 = 1

The given expression becomes: 0 + 1 + ½ + 1 = 2½ = 2.5

Exponents That Are Fractions

Example: 27¹⁄³

27 raised to the ⅓ power

An exponent that is a fraction can seem confusing. But, such exponents have a simple meaning. The original number is the base. In the above example, the base is 27. The denominator in the fraction exponent refers to the root. And the numerator in the fraction exponent refers to the power.

Usually, only examples that have clean simple roots and powers are used on standardized tests.

27 raised to the ⅓ power means:

1) Find the cube root (because the denominator is 3) of 27.

∛27 = 3

2) Then, raise that result to the power of the numerator (our numerator is 1).

3¹ = 3

Therefore, 27¹⁄³ = 3

Or, put in other words, 27 raised to the ⅓ power is equal to 3.

We hope that the above recap of the most common exponent rules that appear on the SHSAT and SAT will help you make your way through those tests. With a little practice, we know taht you will be able to breeze through such questions that test your knowledge of exponents.