Before you read

Answer first — the lesson below explains it.

A biologist models a bacteria population as $P = x^{4} \cdot x^{5}$ cells after $x$ hours. Which expression is equivalent to $P$ ?

A
$x^{9}$
B
$x^{20}$
C
$2 x^{9}$
D
$x^{45}$

Check your answer

Answer:

Before you read

Answer first — the lesson below explains it.

The ratio of a planet's mass to a moon's mass can be expressed as $\frac{a ^{7}}{a ^{3}}$ . Which simplified expression represents this ratio?

A
$a^{4}$
B
$a^{10}$
C
$a^{\frac{7}{3}}$
D
$a^{21}$

Check your answer

Answer:

Theory

The Exponent Rules

Exponents are a shorthand for repeated multiplication: $a^n = \underbrace{a \cdot a \cdot a \cdots a}_{n \text{ times}}$ .

Here are the seven essential rules you need for the SAT:

Rule	Formula	Example
Product Rule	$a^m \cdot a^n = a^{m+n}$	$x^3 \cdot x^4 = x^7$
Quotient Rule	$\frac{a^m}{a^n} = a^{m-n}$	$\frac{x^5}{x^2} = x^3$
Power Rule	$(a^m)^n = a^{mn}$	$(x^3)^2 = x^6$
Product to Power	$(ab)^n = a^n b^n$	$(2x)^3 = 8x^3$
Quotient to Power	$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$	$\left(\frac{x}{3}\right)^2 = \frac{x^2}{9}$
Zero Exponent	$a^0 = 1$ (if $a \neq 0$ )	$5^0 = 1$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$x^{-2} = \frac{1}{x^2}$

$x^3 \cdot x^4 = x^7$ add exponents
$(x^3)^2 = x^6$ multiply exponents
$x^5 / x^2 = x^3$ subtract exponents

The three key exponent rules

Quick check

You just read it. Can you apply it?

A square painting has side length $x^{2}$ inches. If the painting is enlarged so that each side is raised to the 5th power of its original length, the new side length is $(x^{2})^{5}$ . What is this equivalent to?

A
$x^{7}$
B
$x^{10}$
C
$x^{25}$
D
$5 x^{2}$

Check your answer

Answer:

Power rule: $(x^{2})^{5} = x^{2 \times 5} = x^{10}$ .

Theory

Combining Rules

The SAT loves to combine multiple rules in one problem. The key: work step by step and apply one rule at a time.

Example 1

Simplify $\frac{x^5 \cdot x^3}{x^2}$

Numerator: $x^5 \cdot x^3 = x^{5+3} = x^8$ (product rule)

Division: $\frac{x^8}{x^2} = x^{8-2} = x^6$ (quotient rule)

x^6

Example 2

Simplify $(2x^3)^4$

Apply power to both factors: $2^4 \cdot (x^3)^4$

$= 16 \cdot x^{12}$

$= 16x^{12}$

16x^{12}

Example 3

Simplify $\frac{12x^5y^{-2}}{4x^{-1}y^3}$

Coefficients: $\frac{12}{4} = 3$

$x$ -terms: $x^{5-(-1)} = x^{5+1} = x^6$

$y$ -terms: $y^{-2-3} = y^{-5} = \frac{1}{y^5}$

Result: $\frac{3x^6}{y^5}$

\frac{3x^6}{y^5}

Quick check

You just read it. Can you apply it?

In a physics formula, a measurement involves $3^{- 2}$ . What is this value?

A
$- 9$
B
$- 6$
C
$\frac{1}{9}$
D
$\frac{1}{6}$

Check your answer

Answer:

$3^{- 2} = \frac{1}{3 ^{2}} = \frac{1}{9}$ . A negative exponent means take the reciprocal.

Theory

Rational (Fractional) Exponents

Fractional exponents connect exponents to roots:

$a^{\frac{1}{n}} = \sqrt[n]{a} \qquad \text{and} \qquad a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Examples:
- $x^{\frac{1}{2}} = \sqrt{x}$
- $8^{\frac{1}{3}} = \sqrt[3]{8} = 2$
- $x^{\frac{3}{2}} = (\sqrt{x})^3 = x\sqrt{x}$
- $27^{\frac{2}{3}} = (\sqrt[3]{27})^2 = 3^2 = 9$

Quick check

You just read it. Can you apply it?

The volume of a cube with side $3 x^{2}$ is $(3 x^{2})^{3}$ . What is this expression simplified?

A
$9 x^{6}$
B
$27 x^{6}$
C
$3 x^{6}$
D
$27 x^{5}$

Check your answer

Answer:

$(3 x^{2})^{3} = 3^{3} \cdot (x^{2})^{3} = 27 \cdot x^{6} = 27 x^{6}$ .

Common Mistakes

Common Mistakes to Avoid

Adding exponents when bases are different

$x^2 \cdot y^3 = (xy)^5$ $\leftarrow$ WRONG!

The product rule only works when the bases are the same:

x^2 \cdot y^3

cannot be simplified further.

Multiplying exponents instead of adding (product rule)

$x^3 \cdot x^4 = x^{12}$ $\leftarrow$ WRONG!

When multiplying same base: add exponents.

x^3 \cdot x^4 = x^7

Thinking

a^0 = 0

$5^0 = 0$ $\leftarrow$ WRONG!

a^0 = 1

for any

a \neq 0

. Zero exponent means 1, not 0.

Mishandling negative exponents

$2^{-3} = -8$ $\leftarrow$ WRONG!

2^{-3} = \frac{1}{2^3} = \frac{1}{8}

. Negative exponent means reciprocal, not negative number.

Quick check

You just read it. Can you apply it?

A chemistry formula simplifies to $\frac{x ^{3} \cdot x ^{- 5}}{x ^{- 4}}$ . Which expression is equivalent?

A
$x^{2}$
B
$x^{- 6}$
C
$\frac{1}{x ^{2}}$
D
$x^{12}$

Check your answer

Answer:

Numerator: $x^{3} \cdot x^{- 5} = x^{3 + (- 5)} = x^{- 2}$ .
Division: $\frac{x ^{- 2}}{x ^{- 4}} = x^{- 2 - (- 4)} = x^{- 2 + 4} = x^{2}$ .

Strategy

SAT Strategy Tip

Exponent questions appear on every SAT. They range from easy (direct rule application) to hard (combining 3+ rules). The fastest approach: simplify one rule at a time, working from the inside out for nested exponents. When the answer choices use different forms (e.g., $x^{-2}$ vs $\frac{1}{x^2}$ ), convert everything to the same form before comparing.

After reading

Same question as before. What do you say now?

A biologist models a bacteria population as $P = x^{4} \cdot x^{5}$ cells after $x$ hours. Which expression is equivalent to $P$ ?

A
$x^{9}$
B
$x^{20}$
C
$2 x^{9}$
D
$x^{45}$

Check your answer

Answer:

Using the product rule for exponents: $x^{4} \cdot x^{5} = x^{4 + 5} = x^{9}$ .

After reading

Same question as before. What do you say now?

The ratio of a planet's mass to a moon's mass can be expressed as $\frac{a ^{7}}{a ^{3}}$ . Which simplified expression represents this ratio?

A
$a^{4}$
B
$a^{10}$
C
$a^{\frac{7}{3}}$
D
$a^{21}$

Check your answer

Answer:

Quotient rule: $\frac{a ^{7}}{a ^{3}} = a^{7 - 3} = a^{4}$ .

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