Equivalent Expressions Pattern - Exponent Rules

Exponent and Radical Rules

This pattern tests whether you can simplify expressions using exponent laws. The SAT covers the product rule, quotient rule, power-of-a-power rule, negative exponents, and conversions between radicals and fractional exponents. Harder questions chain multiple rules together and ask you to solve for an unknown exponent.

The Core Rules

Product rule: $x^m \cdot x^n = x^{m+n}$ — same base, add exponents
Quotient rule: $\dfrac{x^m}{x^n} = x^{m-n}$ — same base, subtract exponents
Power of a power: $(x^m)^n = x^{mn}$ — multiply exponents
Negative exponent: $x^{-n} = \dfrac{1}{x^n}$ — flip to the denominator
Fractional exponent ↔ radical: $x^{1/n} = \sqrt[n]{x}$ and $x^{m/n} = \sqrt[n]{x^m}$

Worked Examples

Example 1. Which expression is equivalent to $(a^2 b^{-2} c^5)(a b^6 c^2)$, where $a$, $b$, and $c$ are positive?

A) $a^2 b^{-12} c^{10}$
B) $a^2 b^4 c^7$
C) $a^3 b^4 c^7$
D) $a^3 b^{-8} c^7$

Apply the product rule to each base separately:
$a$: $a^2 \cdot a^1 = a^{2+1} = a^3$
$b$: $b^{-2} \cdot b^6 = b^{-2+6} = b^4$
$c$: $c^5 \cdot c^2 = c^{5+2} = c^7$
Result: $a^3 b^4 c^7$
Gotcha: Option A multiplies exponents ($2 \times 1 = 2$) instead of adding. Option B misses the implicit exponent of $1$ on the standalone $a$. Option D subtracts $b$'s exponents instead of adding ($-2 - 6 = -8$).
The answer is C.

Example 2. Which expression is equivalent to $a^{1/9}$, where $a > 0$?

A) $9a$
B) $a^9$
C) $\sqrt[9]{a}$
D) $\dfrac{1}{9}a$

The rule $x^{1/n} = \sqrt[n]{x}$ converts a fractional exponent to a radical.
$a^{1/9} = \sqrt[9]{a}$
Gotcha: Option A treats the exponent as a coefficient ($9 \times a$). Option B flips the fraction to get $a^9$. A fractional exponent $\dfrac{1}{n}$ means the $n$th root, not the $n$th power.
The answer is C.

Example 3. Which expression is equivalent to $\dfrac{a^6 b^{18}}{a^{12} b^6}$, for all $a > 0$ and $b > 0$?

A) $a^6 b^{12}$
B) $\dfrac{b^{12}}{a^6}$
C) $\dfrac{b^3}{a^2}$
D) $a^2 b^3$

Apply the quotient rule:
$a$: $a^{6-12} = a^{-6}$
$b$: $b^{18-6} = b^{12}$
Result: $a^{-6} b^{12} = \dfrac{b^{12}}{a^6}$
Gotcha: Option A ignores the negative exponent and places $a^6$ in the numerator. Option C divides exponents ($18 \div 6 = 3$, $12 \div 6 = 2$) instead of subtracting. With the quotient rule, you always subtract.
The answer is B.

Example 4. Which expression is equivalent to $\dfrac{v^{20} t^8}{v^5 t^{16}}$, assuming $v > 0$ and $t > 0$?

A) $\dfrac{v^{15}}{t^8}$
B) $\dfrac{v^4}{t^2}$
C) $v^4 t^2$
D) $v^{15} t^8$

$v$: $v^{20-5} = v^{15}$
$t$: $t^{8-16} = t^{-8} = \dfrac{1}{t^8}$
Result: $\dfrac{v^{15}}{t^8}$
Gotcha: Option B divides exponents ($20 \div 5 = 4$). Option D ignores the negative exponent on $t$. When the numerator exponent is smaller, you get a negative exponent, which moves that variable to the denominator.
The answer is A.

Example 5. Assuming $m$ and $n$ are positive, which expression is equivalent to $\sqrt[5]{m^8 n^8}$?

A) $(mn)^{5/8}$
B) $(mn)^{8/5}$
C) $(mn)^{40}$
D) $(mn)^{13}$

First, combine the radicand using the power-of-a-product rule: $m^8 n^8 = (mn)^8$.
Then convert the radical to a fractional exponent: $\sqrt[5]{(mn)^8} = (mn)^{8/5}$.
Gotcha: Option A swaps numerator and denominator in the fraction. Option C multiplies ($8 \times 5 = 40$). Option D adds ($8 + 5 = 13$). The $n$th root becomes a denominator of $n$ in the fractional exponent.
The answer is B.

Example 6. The expression $\sqrt[3]{15a} \cdot \left(\sqrt[4]{15a}\right)^2$ is equivalent to $(15a)^{12x}$ for $a > 1$. What is the value of $x$?

Convert to fractional exponents:
$\sqrt[3]{15a} = (15a)^{1/3}$
$\left(\sqrt[4]{15a}\right)^2 = \left((15a)^{1/4}\right)^2 = (15a)^{2/4} = (15a)^{1/2}$
Apply the product rule: $(15a)^{1/3} \cdot (15a)^{1/2} = (15a)^{1/3 + 1/2}$
Add the fractions: $\dfrac{1}{3} + \dfrac{1}{2} = \dfrac{2}{6} + \dfrac{3}{6} = \dfrac{5}{6}$
So the expression equals $(15a)^{5/6}$.
Set equal: $\dfrac{5}{6} = 12x$, so $x = \dfrac{5}{6 \times 12} = \dfrac{5}{72}$
The answer is $\dfrac{5}{72}$ (or $\approx 0.0694$).

Example 7. If $z > 1$, for what value of $x$ is $\dfrac{\sqrt{8z}}{\left(\sqrt[3]{8z}\right)^2}$ equivalent to $(8z)^{10x}$?

Numerator: $\sqrt{8z} = (8z)^{1/2}$
Denominator: $\left(\sqrt[3]{8z}\right)^2 = (8z)^{2/3}$
Quotient rule: $(8z)^{1/2 - 2/3} = (8z)^{3/6 - 4/6} = (8z)^{-1/6}$
Set equal: $-\dfrac{1}{6} = 10x$, so $x = -\dfrac{1}{60}$
Gotcha: Watch the sign carefully — the numerator exponent $\dfrac{1}{2}$ is smaller than the denominator exponent $\dfrac{2}{3}$, so the result is negative.
The answer is $-\dfrac{1}{60}$ (or $\approx -0.0167$).

What to Do on Test Day

• Product rule (multiply): Add exponents. $x^3 \cdot x^5 = x^8$
• Quotient rule (divide): Subtract exponents. $\dfrac{x^7}{x^3} = x^4$
• Power of a power: Multiply exponents. $(x^3)^4 = x^{12}$
• Negative exponent: Flip to denominator. $x^{-3} = \dfrac{1}{x^3}$
• Radical ↔ exponent: $\sqrt[n]{x} = x^{1/n}$. The root index becomes the denominator of the fractional exponent.
• Implicit exponents: A variable written alone (like $a$ in $a^2 \cdot a$) has exponent $1$. Don't forget it.
• Common mistake: Multiplying exponents when you should add (or vice versa). Product rule = add. Power of a power = multiply. Keep them straight.
• Fraction arithmetic in exponents: When adding $\dfrac{1}{3} + \dfrac{1}{2}$, find LCD first: $\dfrac{2}{6} + \dfrac{3}{6} = \dfrac{5}{6}$. Sloppy fraction work is the #1 source of errors in hard exponent problems.