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\(x\)
\(n\)

Integer
Exponents

Master the power of repeated multiplication. Learn to read, write, and evaluate expressions with integer exponents: from positive powers to zero and negative exponents.

Pre-Algebra

What Is an Exponent?

An exponent tells you how many times to multiply a number by itself. Instead of writing \( 3 \times 3 \times 3 \times 3 \), we write \( 3^4 \). The number being multiplied is the base, and the exponent (or power) sits above it.

\(3\)
\(4\)
 base exponent
\(=\)
\(3 \times 3 \times 3 \times 3\)
 4 times

Special Powers: Squared and Cubed

Squared: exponent of 2
\( 5^2 = 5 \times 5 = 25 \)
Think of the area of a square with side 5.
Cubed: exponent of 3
\( 5^3 = 5 \times 5 \times 5 = 125 \)
Think of the volume of a cube with side 5.

Why Do Exponents Matter?

Exponents are a shorthand that appears everywhere in math and science: compound interest, area and volume formulas, scientific notation, and computer science. Without exponents, expressions would be enormously long.

Science
The speed of light: \( 3 \times 10^8 \) m/s
Geometry
Area of a circle: \( \pi r^2 \)
Finance
Compound interest: \( P(1+r)^t \)

The Zero Exponent Rule

Any nonzero number raised to the power of zero equals one. This isn't just a definition: it follows from the pattern of dividing by the base each time the exponent decreases.

\(2^4 = 16\)
\(2^3 = 8\)
\(2^2 = 4\)
\(2^1 = 2\)
\(2^0 = 1\)
\(\div 2\)
\(\div 2\)
\(\div 2\)
\(\div 2\)
 Rule
\(a^0 = 1 \ (a \neq 0)\)
 Each step divides by the base.
So the pattern forces \(2^0 = 1\).

Negative Exponents

A negative exponent means "take the reciprocal." Instead of multiplying, you divide. The pattern from the previous slide keeps going below zero.

\(2^0 = 1\)
\(2^{-1} = \frac{1}{2}\)
\(2^{-2} = \frac{1}{4}\)
\(2^{-3} = \frac{1}{8}\)
\(\div 2\)
\(\div 2\)
\(\div 2\)
 Rule
\(a^{-n} = \frac{1}{a^n}\)
 The negative sign flips the base to its reciprocal.

Negative Base vs. Negative Exponent

Students often confuse a negative base with a negative exponent. They do completely different things. Pay close attention to where the negative sign sits.

Expression Meaning Value
\( (-3)^2 \) Negative base, squared +9
\( -3^2 \) Negate after squaring -9
\( 3^{-2} \) Negative exponent: reciprocal \( \frac{1}{9} \)

Product of Powers Rule

When you multiply two powers with the same base, you add the exponents. This works because you're combining the total number of factors.

\(a^m \times a^n = a^{m+n}\)
 Example:
\(2^3 \times 2^4\)
\(=\)
\(2^{3+4}\)
\(=\)
\(2^7 = 128\)

Quotient of Powers Rule

When you divide two powers with the same base, you subtract the exponents. Dividing cancels out common factors.

\(a^m \div a^n = a^{m-n}\)
 Example:
\(5^6 \div 5^2\)
\(=\)
\(5^{6-2}\)
\(=\)
\(5^4 = 625\)

Power of a Power Rule

When you raise a power to another power, you multiply the exponents. You're repeating the repeated multiplication.

\((a^m)^n = a^{m \times n}\)
 Example:
\((3^2)^4\)
\(=\)
\(3^{2 \times 4}\)
\(=\)
\(3^8 = 6561\)

How to Evaluate an Expression

1
Identify the base and exponent. Circle them clearly: the base is the repeated factor, the exponent is the count.
2
Check for special cases. Zero exponent? Negative exponent? Apply the rule before multiplying.
3
Expand if needed. Write out the multiplication: \( 4^3 = 4 \times 4 \times 4 \).
4
Multiply step by step. Go left to right: \( 4 \times 4 = 16 \), then \( 16 \times 4 = 64 \).
5
Double-check sign. A negative base with an even exponent is positive. With an odd exponent, it's negative.

Example 1: Positive Exponent

Evaluate \( (-2)^5 \).

1
Base: \(-2\). Exponent: \(5\). The base is negative and inside parentheses.
2
Expand: \( (-2)^5 = (-2) \times (-2) \times (-2) \times (-2) \times (-2) \)
3
Multiply: \( 4 \times (-2) = -8 \), then \( -8 \times (-2) = 16 \), then \( 16 \times (-2) = -32 \)
4
Sign check: Odd exponent with a negative base gives a negative result.

Example 2: Negative Exponent

Evaluate \( 4^{-3} \).

1
Negative exponent rule: Flip to a fraction. \( 4^{-3} = \frac{1}{4^3} \)
2
Evaluate the denominator: \( 4^3 = 4 \times 4 \times 4 = 64 \)
3
Write the result: \( 4^{-3} = \frac{1}{64} \)

Example 3: Using the Rules Together

Simplify \( \frac{3^5 \times 3^{-2}}{3^2} \).

1
Numerator: Product rule: \( 3^5 \times 3^{-2} = 3^{5+(-2)} = 3^3 \)
2
Full expression: Quotient rule: \( \frac{3^3}{3^2} = 3^{3-2} = 3^1 \)
3
Evaluate: \( 3^1 = 3 \)

Common Mistakes to Avoid

✕ Multiplying Base by Exponent
Writing \( 2^5 = 10 \) instead of \( 2^5 = 32 \). The exponent tells how many times to multiply, not to multiply base times exponent.
✕ Negative Exponent = Negative
Thinking \( 5^{-2} = -25 \). Actually, \( 5^{-2} = \frac{1}{25} \), which is positive. The negative exponent flips, not negates.
✕ Forgetting Parentheses
Confusing \( (-4)^2 = 16 \) with \( -4^2 = -16 \). Without parentheses, only the 4 is squared.

Exponent Rules Summary

Rule Formula Example
Product of Powers \( a^m \times a^n = a^{m+n} \) \( 2^3 \times 2^4 = 2^7 \)
Quotient of Powers \( a^m \div a^n = a^{m-n} \) \( 5^6 \div 5^2 = 5^4 \)
Power of a Power \( (a^m)^n = a^{m \times n} \) \( (3^2)^4 = 3^8 \)
Zero Exponent \( a^0 = 1 \) \( 7^0 = 1 \)
Negative Exponent \( a^{-n} = \frac{1}{a^n} \) \( 4^{-2} = \frac{1}{16} \)

Question 1 of 5

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

Question 2 of 5

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

Question 3 of 5

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

Question 4 of 5

Simplify: \( 5^3 \times 5^2 \)

Question 5 of 5

What is \( (-3)^3 \)?