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\(x\) \(n\)

Exponent
Expressions

Master the language of repeated multiplication. Learn to read, write, and evaluate expressions with exponents: the key to unlocking algebra.

SSAT Middle Level: Quantitative

What Is an Exponent?

An exponent tells you how many times to multiply a number by itself. Instead of writing the same factor over and over, we use a small raised number as a shorthand.

\( 3^4 \) Base Exponent = \( 3 \times 3 \times 3 \times 3 \) \( = 81 \)

Reading Exponent Notation

Exponents have special names you will see on the SSAT. Knowing how to say them out loud helps you understand what they mean.

\( 5^2 \)
"Five squared" : \( 5 \times 5 = 25 \)
\( 2^3 \)
"Two cubed" : \( 2 \times 2 \times 2 = 8 \)
\( 4^5 \)
"Four to the fifth power" : \( 4^5 = 1024 \)

Why Exponents Matter

Exponents aren't just a shorthand trick: they appear across science, money, and everyday math. Here's why they matter on the SSAT and beyond.

Area
The area of a square with side \( s \) is \( s^2 \).
Volume
The volume of a cube with edge \( e \) is \( e^3 \).
Growth
Doubling every day: day \( n \) gives \( 2^n \) items.

Core Rule: Evaluating a Power

\(2\) \(\times\) \(2\) \(\times\) \(2\) \(\times\) \(2\) \(\times\) \(2\) 5 factors of 2 \( 2^5 \) = \( 32 \)

Special Exponents: \(a^1\) and \(a^0\)

Exponent of 1
Any number to the first power is itself: \( a^1 = a \).
Example: \( 7^1 = 7 \)
Exponent of 0
Any nonzero number to the zero power equals 1: \( a^0 = 1 \).
Example: \( 5^0 = 1 \)

Exponents vs. Multiplication

Students often confuse repeated multiplication (exponents) with repeated addition (multiplication). Let's see the difference side by side.

Exponent Multiplication
Expression \( 3^4 \) \( 3 \times 4 \)
Meaning \( 3\times3\times3\times3 \) \( 3+3+3+3 \)
Result 81 12
Growth Very fast (exponential) Moderate (linear)

How to Evaluate Step by Step

1
Identify the base and the exponent.
2
Write out the base as many times as the exponent says.
3
Multiply from left to right, one pair at a time.
4
Check special cases: Is the exponent 0 or 1? Apply the rule immediately.

Example 1: Evaluate \( 4^3 \)

Step 1: Identify \(\text{Base} = 4, \text{Exponent} = 3\) Step 2: Expand \( 4 \times 4 \times 4 \) Step 3: Multiply \( 4 \times 4 = 16 \) then \( 16 \times 4 = 64 \) Answer: \( 4^3 = 64 \)

Example 2: Evaluate \( 10^4 \)

Step 1: Identify \(\text{Base} = 10, \text{Exponent} = 4\) Step 2: Expand \( 10 \times 10 \times 10 \times 10 \) Shortcut for Powers of 10 Write a \(1\) followed by \(4\) zeros: \(10{,}000\) Answer: \( 10^4 = 10{,}000 \)

Example 3: Evaluate \( 2^3 + 5^2 \)

1
Handle each exponent separately first (order of operations: exponents before addition).
2
Evaluate \( 2^3 \): \( 2 \times 2 \times 2 = \) 8
3
Evaluate \( 5^2 \): \( 5 \times 5 = \) 25
4
Add the results: 8 + 25 = 33

Parentheses Matter: \( (-3)^2 \) vs. \( -3^2 \)

\( (-3)^2 \) \( -3^2 \)
Meaning \( (-3)\times(-3) \) \( -(3\times3) \)
Calculation Neg × Neg = Positive Square first, then negate
Result +9 −9

Powers to Memorize

These perfect squares and cubes appear repeatedly on the SSAT. Knowing them by heart saves valuable time.

Perfect Squares Value Perfect Cubes Value
\( 1^2 \)1\( 1^3 \)1
\( 2^2 \)4\( 2^3 \)8
\( 3^2 \)9\( 3^3 \)27
\( 4^2 \)16\( 4^3 \)64
\( 5^2 \)25\( 5^3 \)125
\( 6^2 \)36\( 10^3 \)1,000
\( 10^2 \)100
\( 12^2 \)144

Exponents in Larger Expressions

On the SSAT, exponents often appear inside bigger expressions. Always follow the order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

1
Parentheses first: simplify anything inside \( ( \, ) \).
2
Exponents next: evaluate all powers.
3
Multiply/Divide from left to right.
4
Add/Subtract from left to right.

Common Mistakes to Avoid

✕ Multiplying Base by Exponent
Wrong: \( 3^4 = 12 \). That's \( 3 \times 4 \). Correct: \( 3^4 = 81 \).
✕ Forgetting Parentheses
Wrong: \( -5^2 = 25 \). Correct: \( -5^2 = -25 \). You need \( (-5)^2 \) to get \( 25 \).
✕ Zero Power Confusion
Wrong: \( 7^0 = 0 \). Correct: \( 7^0 = 1 \). Any nonzero base to the 0 power is 1.

Exponent Rules Summary

Rule Formula Example
Basic Power \( a^n = a \times a \times \cdots \) \( 5^3 = 125 \)
Exponent of 1 \( a^1 = a \) \( 9^1 = 9 \)
Exponent of 0 \( a^0 = 1 \) (if \( a \neq 0 \)) \( 6^0 = 1 \)
Powers of 10 \( 10^n = 1\underbrace{00\ldots0}_{n} \) \( 10^5 = 100{,}000 \)
Negative Base (Parentheses) \( (-a)^2 = a^2 \) vs \( -a^2 \) \( (-3)^2 = 9 \), but \( -3^2 = -9 \)

Question 1 of 5

What is the value of \( 6^3 \)?

Question 2 of 5

What is the value of \( 10^0 + 10^1 \)?

Question 3 of 5

Which expression has the greatest value?

Question 4 of 5

What is the value of \( 2 \times 3^2 + 4 \)?

Question 5 of 5

What is the value of \( (-4)^2 - 4^2 \)?