Whole Number Exponents

What is an Exponent?

An exponent is a shorthand way of writing repeated multiplication. When you see an expression like $a^n$, it means you multiply the number $a$ by itself $n$ times.

• Base ($a$): The number being multiplied.
• Exponent ($n$): The small number written high and to the right, telling you how many times to use the base as a factor.

Writing Repeated Multiplication

Instead of writing out a long multiplication problem, you can use an exponent.

Example: Write $8 \times 8 \times 8$ using an exponent.

• The number being multiplied is $8$, so the base is $8$.
• It is multiplied by itself $3$ times, so the exponent is $3$.
• Answer: $8^3$ (read as "eight to the third power").

Evaluating Exponents

To evaluate (or solve) an expression with an exponent, simply write it out as repeated multiplication and find the product.

Example: Calculate $3^4$

• This means $3$ multiplied by itself $4$ times: $3 \times 3 \times 3 \times 3$
• Let's multiply step-by-step:
  • $3 \times 3 = 9$
  • $9 \times 3 = 27$
  • $27 \times 3 = 81$
• Answer: $81$

Exponents with Fractions

You can also use exponents with fractions. The rule is exactly the same: multiply the fraction by itself!

Example: Evaluate $\left(\frac{2}{3}\right)^2$

• This means $\frac{2}{3} \times \frac{2}{3}$.
• Multiply the numerators and the denominators: $\frac{2 \times 2}{3 \times 3} = \frac{4}{9}$
• Answer: $\frac{4}{9}$

Watch Out for Common Mistakes!

A very common mistake is multiplying the base by the exponent.

• $3^4$ is not $3 \times 4 = 12$.
• $3^4$ is $3 \times 3 \times 3 \times 3 = 81$.

Always remember: the exponent tells you how many times to write the base in a multiplication problem.