Exponent Laws
Special Cases & Number Sense
GMAT Pattern: When comparing $2^{10}$ vs $10^2 = 100$ vs $2^{10} = 1024$, large exponents on small bases beat small exponents on large bases.
Roots & Radicals
GMAT Simplification Tactics
GMAT problems involving exponents often respond to: (1) expressing everything in terms of a common base, (2) factoring out common powers, or (3) looking for patterns in unit digits.
10 Exponent Precision Traps
1. $(a+b)^2 \neq a^2 + b^2$
$(a+b)^2 = a^2 + 2ab + b^2$. The middle term is always forgotten.
2. Negative base with even/odd exponent
$(-2)^3 = -8$ but $(-2)^4 = +16$. Sign depends on parity of exponent.
3. Square root is always non-negative
$\sqrt{9} = 3$, not ±3. The radical symbol returns the principal (positive) root.
4. $x^2 = 9$ has two solutions
$x^2 = 9$ → $x = 3$ or $x = -3$. NEVER just one solution.
5. Adding exponents instead of multiplying
$2^3 \times 2^4 = 2^7$, not $2^{12}$. Add exponents when MULTIPLYING same base.
6. Multiplying exponents when adding bases
$2^3 + 2^4 \neq 2^7$. You can only simplify by factoring: $2^3(1+2) = 3 \times 2^3 = 24$.
7. Fractional exponent confusion
$8^{2/3} = (8^{1/3})^2 = 2^2 = 4$. Denominator = root, numerator = power.
8. Zero exponent edge case
$0^0$ is undefined — the GMAT will never ask you to evaluate it, but be aware in DS.
9. Inequality with even exponents
$x^2 > 9$ does NOT mean $x > 3$. It means $x > 3$ OR $x < -3$.
10. Comparing $a^b$ and $b^a$
The larger base wins when both are large, but $2^3 < 3^2$ shows the pattern can reverse for small bases.
10 GMAT Practice Questions
What is the value of $\dfrac{3^8 - 3^6}{3^5}$?
(C) 54. Factor numerator: $3^6(3^2 - 1) = 3^6 \times 8$. Divide by $3^5$: $3 \times 8 = 24$. Wait — $3^6 \times 8 ÷ 3^5 = 3^1 \times 8 = 24$. Answer is (B) 24. (B) is correct.
If $2^x \cdot 4^3 = 8^4$, what is $x$?
(C) 6. Convert to base 2: $2^x \cdot 2^6 = 2^{12}$ → $2^{x+6} = 2^{12}$ → $x+6 = 12$ → $x = 6$.
What is $\sqrt{72} + \sqrt{50}$?
(C) $11\sqrt{2}$. $\sqrt{72} = 6\sqrt{2}$ and $\sqrt{50} = 5\sqrt{2}$. Sum = $11\sqrt{2}$.
Which is greater: $3^4$ or $4^3$?
(A) $3^4$. $3^4 = 81$ and $4^3 = 64$. So $3^4 > 4^3$.
If $x^2 = 25$ and $y^2 = 49$, what is the maximum value of $x - y$?
(C) 12. $x = \pm5$, $y = \pm7$. Maximum $x - y$ = max $x$ − min $y$ = $5 - (-7) = 12$.
What is the value of $(-2)^3 + (-3)^2 - (-1)^5$?
(C) 2. $(-2)^3 = -8$, $(-3)^2 = 9$, $(-1)^5 = -1$. Expression = $-8 + 9 - (-1) = -8 + 9 + 1 = 2$.
Is $n^2$ an even integer?
(1) $n$ is even
(2) $n^3$ is even
(D) EACH alone sufficient. (1): If $n$ is even, $n^2$ is even. Sufficient. (2): $n^3$ is even only if $n$ is even (odd³ = odd). So $n$ is even → $n^2$ is even. Sufficient. Both (1) and (2) alone are sufficient → answer (D).
What is the value of $\dfrac{(2^4)^3}{2^9}$?
(C) 8. $(2^4)^3 = 2^{12}$. $\frac{2^{12}}{2^9} = 2^3 = 8$.
If $a = 2^{10}$ and $b = 10^3$, which of the following is true?
(C) $a > b$. $a = 2^{10} = 1024$ and $b = 10^3 = 1000$. So $a > b$.
Simplify: $\sqrt{\dfrac{144}{25}}$
(B) $\frac{12}{5}$. $\sqrt{\frac{144}{25}} = \frac{\sqrt{144}}{\sqrt{25}} = \frac{12}{5}$.
Lesson Summary — Key Takeaways
7 laws — memorize all
Product: add. Quotient: subtract. Power: multiply. Zero: 1. Negative: flip.
Common base is the master tactic
Any product/quotient of powers of 2, 4, 8 → express all in base 2 first.
√ always returns positive
$\sqrt{9} = 3$, not ±3. But $x^2 = 9$ gives $x = ±3$.
$(a+b)^2 ≠ a^2+b^2$
Always expand fully: $(a+b)^2 = a^2 + 2ab + b^2$. The $2ab$ term is crucial.