AMC 10 · 2003 · #9

Grade 8 algebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

exponentsfraction-arithmetic identify-subproblemspattern-recognition ↑ Prerequisites: exponents

📏 Medium solution 💡 2 insights

Problem

The expression

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

stacks three cube roots around one innermost square root, with one extra factor of x inside every cube root. Taking x greater than or equal to 0 so that every root is defined, simplify the whole tower into a single root of x.

Pick an answer.

(A)

$\sqrt{x}$

(B)

$\sqrt[3]{x^{2}}$

(C)

$\sqrt[27]{x^{2}}$

(D)

$\sqrt[54]{x}$

(E)

$\sqrt[81]{x^{80}}$

AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

A four-layer nested radical is too much to simplify in one move, so tool #7 (Identify Subproblems) says: peel it from the inside out, treating each cube root as its own small job. The key move that makes every layer easy is rewriting roots as fractional powers, because then "multiply by x" just adds 1 to an exponent and "cube root" just divides an exponent by 3. Tool #5 (Look for a Pattern) pays off almost immediately: the innermost layer collapses to x¹/2, and every layer after it does exactly the same thing, so you never have to redo the work. Tool #6 (Guess and Check) gives an independent confirmation — putting in a perfect square like x=64 and simplifying by hand should return √(x).

1STEP 1

Rewrite every root as a power

Roots are fractional powers: $\sqrt{x}=x^{1/2}$ and a cube root is the $\frac{1}{3}$ power, so multiplying by x adds 1 to the exponent.

√(x)=x¹/2, ∛(y)=y¹/3

2STEP 2

Simplify the innermost layer

Innermost: $x\sqrt{x}=x^{3/2}$ , and its cube root is $(x^{3/2})^{1/3}=$ $x^{1/2}$ .

x√(x)=x¹ · x¹/2=x³/2 → ∛(x³/2)=x^(3/2)(1/3)=x¹/2

3STEP 3

Notice the layer repeats

The next layer repeats it: times x gives $x^{3/2}$ , the cube root returns $x^{1/2}$ — since $\frac{3}{2}\cdot\frac{1}{3}=\frac{1}{2}$ , every layer is a fixed point.

x · x¹/2=x³/2 → (x³/2)¹/3=x^(3/2)(1/3)=x¹/2

4STEP 4

Apply the outermost layer

One layer left: $x\cdot x^{1/2}=x^{3/2}$ , and the last cube root gives $x^{1/2}$ , the square root of x — choice (A).

∛(x · x¹/2)=∛(x³/2)=x¹/2=√(x) → (A)

Answer

√(x)

Look at the scale of the choices. The only operations here are adding 1 to an exponent and dividing an exponent by 3, starting from small numbers, so the exponent can never grow into things like

\frac{1}{54}

\frac{80}{81}

— that rules out (C), (D), and (E) on size alone. The exponent oscillates between

\frac{3}{2}

and

\frac{1}{2}

, and because the tower ends on a cube root it lands on

\frac{1}{2}

, i.e.

\sqrt{x}

— matching choice (A) and not (B)'s

x^{2/3}

💡Key takeaway

Turn roots into fractional powers, then simplify a nested radical from the inside out — multiplying by x adds 1 to the exponent and a cube root divides it by 3.

Rewrite every root as a power
Simplify the innermost layer
Notice the layer repeats
Apply the outermost layer

Rewrite every root as a power

Simplify the innermost layer

Notice the layer repeats

Apply the outermost layer

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