Exponent
Expressions
Master the rules of exponents to simplify, evaluate, and compare expressions with confidence on the SSAT.
SSAT Upper Level: Quantitative
What Is an Exponent?
An exponent tells you how many times to multiply a number (the base) by itself. The expression \( a^n \) means multiply \( a \) by itself \( n \) times.
Anatomy of Exponent Expressions
Base
The number being multiplied. In \( 5^3 \), the base is 5.
Exponent
How many times to multiply. In \( 5^3 \), the exponent is 3.
Power
The entire expression \( 5^3 \) or its value, 125.
Why Exponents Matter
Exponents appear throughout the SSAT in arithmetic, algebra, and word problems. Knowing the rules lets you simplify complex expressions quickly, without a calculator.
The Core Exponent Rules
1
Product Rule: \( a^m \cdot a^n = a^{m+n} \)
2
Quotient Rule: \( \dfrac{a^m}{a^n} = a^{m-n} \)
3
Power Rule: \( (a^m)^n = a^{m \cdot n} \)
4
Zero Exponent: \( a^0 = 1 \) (where \( a \neq 0 \))
5
Negative Exponent: \( a^{-n} = \dfrac{1}{a^n} \)
Product and Quotient Rules
Power Rule and Special Cases
Power of a Power
\( (a^m)^n = a^{m \cdot n} \)
Example: \( (2^3)^2 = 2^6 = \) 64
Example: \( (2^3)^2 = 2^6 = \) 64
Power of a Product
\( (ab)^n = a^n \cdot b^n \)
Example: \( (3 \cdot 4)^2 = 3^2 \cdot 4^2 = 9 \cdot 16 = \) 144
Example: \( (3 \cdot 4)^2 = 3^2 \cdot 4^2 = 9 \cdot 16 = \) 144
Power of a Quotient
\( \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n} \)
Example: \( \left(\dfrac{2}{3}\right)^3 = \dfrac{8}{27} \)
Example: \( \left(\dfrac{2}{3}\right)^3 = \dfrac{8}{27} \)
Zero and Negative Exponents
Negative Base vs Negative Exponent
| Negative Base | Negative Exponent | |
|---|---|---|
| Form | \( (-a)^n \) | \( a^{-n} \) |
| Example | \( (-2)^3 = -8 \) | \( 2^{-3} = \frac{1}{8} \) |
| What changes? | The sign of the result | The result becomes a fraction |
| Even exponent | \( (-2)^4 = +16 \) | \( 2^{-4} = \frac{1}{16} \) |
| Odd exponent | \( (-2)^3 = -8 \) | \( 2^{-3} = \frac{1}{8} \) |
How to Evaluate Step by Step
1
Identify the base and exponent. Check for parentheses and negative signs carefully.
2
Apply exponent rules. Simplify using product, quotient, or power rules before computing.
3
Handle special exponents. Zero exponent → 1. Negative exponent → flip to a fraction.
4
Compute the final value. Multiply out or simplify the fraction completely.
5
Check the sign. Negative base with an odd exponent → negative result. Even exponent → positive.
Example 1: Product Rule
Evaluate \( 2^3 \cdot 2^4 \).
1
Same base? Yes, both bases are 2.
2
Apply product rule: \( 2^3 \cdot 2^4 = 2^{3+4} = 2^7 \)
3
Compute: \( 2^7 = 128 \)
Example 2: Negative Exponents
Evaluate \( \dfrac{3^5}{3^7} \).
1
Same base? Yes, both bases are 3.
2
Apply quotient rule: \( \dfrac{3^5}{3^7} = 3^{5-7} = 3^{-2} \)
3
Handle negative exponent: \( 3^{-2} = \dfrac{1}{3^2} = \dfrac{1}{9} \)
Example 3: Combined Rules
Evaluate \( \dfrac{(2^3)^2 \cdot 2^{-4}}{2^0} \).
1
Power rule on numerator: \( (2^3)^2 = 2^{6} \)
2
Product rule: \( 2^{6} \cdot 2^{-4} = 2^{6+(-4)} = 2^{2} \)
3
Denominator: \( 2^0 = 1 \)
4
Final: \( \dfrac{2^2}{1} = 4 \)
Example 4: Negative Base Pitfall
Evaluate \( -4^2 \) and \( (-4)^2 \).
\( -4^2 \)
Only 4 is squared:
\( -(4^2) = -(16) = \) -16
\( -(4^2) = -(16) = \) -16
\( (-4)^2 \)
The entire \( -4 \) is squared:
\( (-4)(-4) = \) +16
\( (-4)(-4) = \) +16
Common Mistakes
✕ Multiplying Exponents
Writing \( 2^3 \cdot 2^4 = 2^{12} \). You should add exponents: \( 2^{3+4} = 2^7 \).
✕ Negative = Negative
Thinking \( 3^{-2} = -9 \). A negative exponent means a fraction: \( \frac{1}{9} \).
✕ Ignoring Parentheses
Treating \( -5^2 \) and \( (-5)^2 \) as the same. They give \( -25 \) and \( +25 \), respectively.
Exponent Rules Summary
| Rule | Formula | Example |
|---|---|---|
| Product | \( a^m \cdot a^n = a^{m+n} \) | \( 5^2 \cdot 5^3 = 5^5 \) |
| Quotient | \( \frac{a^m}{a^n} = a^{m-n} \) | \( \frac{7^4}{7^2} = 7^2 \) |
| Power | \( (a^m)^n = a^{mn} \) | \( (3^2)^4 = 3^8 \) |
| Zero | \( a^0 = 1 \) | \( 99^0 = 1 \) |
| Negative | \( a^{-n} = \frac{1}{a^n} \) | \( 4^{-2} = \frac{1}{16} \) |
| Product Power | \( (ab)^n = a^n b^n \) | \( (2 \cdot 5)^3 = 2^3 \cdot 5^3 \) |
Question 1 of 5
What is the value of \( 3^2 \cdot 3^3 \)?
Using the product rule: \( 3^2 \cdot 3^3 = 3^{2+3} = 3^5 = 243 \). Add the exponents when multiplying with the same base.
Question 2 of 5
What is the value of \( 5^{-2} \)?
A negative exponent means take the reciprocal: \( 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \). It does not make the answer negative.
Question 3 of 5
What is the value of \( (-3)^4 \)?
The base \( -3 \) is inside parentheses, so the entire \( -3 \) is raised to the 4th power. Since the exponent is even, the result is positive: \( (-3)^4 = 81 \).
Question 4 of 5
Simplify: \( \dfrac{(2^4)^3}{2^{10}} \)
Apply the power rule to the numerator: \( (2^4)^3 = 2^{12} \). Then the quotient rule: \( \frac{2^{12}}{2^{10}} = 2^{12-10} = 2^2 = 4 \).
Question 5 of 5
Which expression is not equal to \( 64 \)?
\( 8^3 = 512 \), not 64. The others all equal 64: \( 2^6 = 64 \), \( 4^3 = 64 \), and \( (2^2)^3 = 2^6 = 64 \).