Exponents and Square Roots

An exponent tells you how many times to multiply a number by itself. A square root reverses the process — it asks “what number, multiplied by itself, gives this result?” These two operations are inverses of each other, like multiplication and division.

Exponents

$a^n = \underbrace{a \times a \times a \times \cdots \times a}_{n \text{ times}}$

• $a$ is the base — the number being multiplied
• $n$ is the exponent (or power) — how many times to multiply

Common Powers

$2^1 = 2 \qquad 2^2 = 4 \qquad 2^3 = 8 \qquad 2^4 = 16 \qquad 2^5 = 32$

$3^1 = 3 \qquad 3^2 = 9 \qquad 3^3 = 27 \qquad 3^4 = 81$

$5^1 = 5 \qquad 5^2 = 25 \qquad 5^3 = 125$

$10^1 = 10 \qquad 10^2 = 100 \qquad 10^3 = 1{,}000 \qquad 10^4 = 10{,}000$

Vocabulary

“Squared” and “cubed” have special names because they connect to geometry — the area of a square with side 4 is $4^2 = 16$, and the volume of a cube with side 4 is $4^3 = 64$.

Special Exponents

$a^1 = a \qquad \text{(any number to the first power is itself)}$

$a^0 = 1 \qquad \text{(any nonzero number to the zero power is 1)}$

Why is $a^0 = 1$? Look at the pattern:

$2^4 = 16, \quad 2^3 = 8, \quad 2^2 = 4, \quad 2^1 = 2, \quad 2^0 = \;?$

Each step divides by 2, so $2^0 = 2 \div 2 = 1$.

Example 1: Evaluate $5^3$

$5^3 = 5 \times 5 \times 5 = 25 \times 5 = 125$

Example 2: Evaluate $2^6$

$2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64$

Example 3: Evaluate $10^5$

$10^5 = 100{,}000$

For powers of 10, the exponent equals the number of zeros.

Square Roots

The square root of a number $n$ is the value that, when multiplied by itself, gives $n$:

$\sqrt{n} = x \quad \text{means} \quad x^2 = n$

Example 4: $\sqrt{25}$

What number squared equals 25? $5 \times 5 = 25$, so $\sqrt{25} = 5$.

Example 5: $\sqrt{144}$

$12 \times 12 = 144$, so $\sqrt{144} = 12$.

Perfect Squares

A perfect square is a number whose square root is a whole number. These are worth memorizing:

Estimating Non-Perfect Square Roots

Most numbers are not perfect squares, so their square roots are not whole numbers. You can estimate by finding the two perfect squares it falls between.

Example 6: Estimate $\sqrt{50}$

$7^2 = 49$ and $8^2 = 64$, so $\sqrt{50}$ is between 7 and 8.

Since 50 is very close to 49, $\sqrt{50} \approx 7.07$.

Example 7: Estimate $\sqrt{200}$

$14^2 = 196$ and $15^2 = 225$, so $\sqrt{200}$ is between 14 and 15.

$200$ is close to 196, so $\sqrt{200} \approx 14.1$.

Cube Roots

A cube root asks “what number cubed gives this result?”

$\sqrt[3]{n} = x \quad \text{means} \quad x^3 = n$

Example 8: $\sqrt[3]{27}$

$3 \times 3 \times 3 = 27$, so $\sqrt[3]{27} = 3$.

Common Cube Roots

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Exponents mean repeated multiplication: $a^n = a$ multiplied by itself $n$ times
• Square roots reverse squaring: $\sqrt{n}$ asks “what squared gives $n$?”
• $a^0 = 1$ for any nonzero $a$; $a^1 = a$ always
• Perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100…) have whole-number square roots
• Estimate non-perfect square roots by finding the two nearest perfect squares
• Cube roots reverse cubing: $\sqrt[3]{n}$ asks “what cubed gives $n$?”

Return to Arithmetic for more foundational math topics.