AMC 10 · 2008 · #3

Grade 8 algebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

exponentsfraction-arithmetic identify-subproblems ↑ Prerequisites: exponents

📏 Medium solution 💡 1 insight

Problem

x

is a positive real number, and the expression

\sqrt[3]{x\sqrt{x}}

wraps a cube root around the product of

x

and its own square root. Rewrite it as a single power of

x

Pick an answer.

(A)

$\ x^{1/6}$

(B)

$\ x^{1/4}$

(C)

$\ x^{3/8}$

(D)

$\ x^{1/2}$

(E)

$\ x$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The expression stacks two radical layers — an inner square root and an outer cube root — over the same base x. Tool #7 (Identify Subproblems) peels them one layer at a time by turning each radical into a fractional exponent, then combining. Tool #6 (Guess and Check) gives an independent confirmation: pick a convenient x=64 and compute both sides numerically. Tool #3 (Eliminate Possibilities) uses that one number to knock out the decoy exponents, since only one choice can produce the same value.

1STEP 1

Turn radicals into fractional exponents

A square root is the $\frac{1}{2}$ power and a cube root the $\frac{1}{3}$ power, so the whole thing is $(x\cdot x^{\frac{1}{2}})^{\frac{1}{3}}$ .

∛(x√(x))=(x · x¹/2)¹/3

2STEP 2

Combine the product inside

Same base means the exponents add: $1+\frac{1}{2}=\frac{3}{2}$ , so the inside collapses to $x^{\frac{3}{2}}$ .

x¹ · x¹/2=x¹+1/2=x³/2

3STEP 3

Apply the outer cube root

A power of a power multiplies the exponents: $\frac{3}{2}\cdot\frac{1}{3}=\frac{1}{2}$ , so the expression equals $x^{\frac{1}{2}}$ .

(x³/2)¹/3=x³/2·1/3=x³/6=x¹/2

4STEP 4

Check by plugging in x=64

Test $x=64$ : $64\sqrt{64}=512$ and $\sqrt[3]{512}=8=\sqrt{64}$ , a value only (D) can produce.

64·√(64)=512, ∛(512)=8=64¹/2 → (D)

Answer

x¹/2

Each root shrinks the exponent: the product inside builds up to x³/2, then the cube root cuts that exponent to a third, landing at x¹/2 — a sensible middle value between x⁰=1 and x¹=x. The numeric test seals it: at x=64 the expression is 8, and 64¹/2=8 while every other choice misses. Choice (A) x¹/6 is the trap for multiplying the three exponents 1·1/2·1/3 instead of adding then multiplying, and (E) x is what you get by dropping the roots altogether.

💡Key takeaway

Turn every root into a fraction-power — a square root is the half power, a cube root is the third power — then add exponents when you multiply and multiply exponents when you stack a power on a power.

Turn radicals into fractional exponents
Combine the product inside
Apply the outer cube root
Check by plugging in x=64

Turn radicals into fractional exponents

Combine the product inside

Apply the outer cube root

Check by plugging in x=64

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