AMC 10 · 2003 · #9
Grade 8 algebraSimplify
3x3x3xx.
Pick an answer.
AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Simplify the nested-radical expression $\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}$ into a single root of $x$, and match it to one of the five answer choices.
Givens: The expression is $\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}$, three cube roots wrapped around one innermost square root; Each layer multiplies by another factor of $x$ before a cube root is taken; Answer choices: (A) $\sqrt{x}$, (B) $\sqrt[3]{x^{2}}$, (C) $\sqrt[27]{x^{2}}$, (D) $\sqrt[54]{x}$, (E) $\sqrt[81]{x^{80}}$
Unknowns: The single simplified root of $x$ that the whole tower equals
Understand
Restated: Simplify the nested-radical expression $\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}$ into a single root of $x$, and match it to one of the five answer choices.
Givens: The expression is $\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}$, three cube roots wrapped around one innermost square root; Each layer multiplies by another factor of $x$ before a cube root is taken; Answer choices: (A) $\sqrt{x}$, (B) $\sqrt[3]{x^{2}}$, (C) $\sqrt[27]{x^{2}}$, (D) $\sqrt[54]{x}$, (E) $\sqrt[81]{x^{80}}$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #5 Look for a Pattern, #6 Guess and Check
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 $x^{1/2}$, 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 $\sqrt{x}$.
Execute — Answer: A
8.EE.A.2 Step 1 Rewrite every root as a power
- Turn each root into a fractional exponent so the operations become arithmetic on exponents.
- A square root is the $\tfrac{1}{2}$ power and a cube root is the $\tfrac{1}{3}$ power: $\sqrt{x}=x^{1/2}$ and $\sqrt[3]{y}=y^{1/3}$.
- From here, multiplying by $x$ adds $1$ to the exponent (product rule), and taking a cube root multiplies the exponent by $\tfrac{1}{3}$ (power-of-a-power rule).
💡 Roots are just fractional powers, so switching to exponents turns a scary tower into simple exponent arithmetic.
8.EE.A.1 Step 2 Simplify the innermost layer
- Start at the very center: $x\sqrt{x}=x^{1}\cdot x^{1/2}=x^{3/2}$ by adding exponents.
- Now take the innermost cube root of that: $\sqrt[3]{x^{3/2}}=\left(x^{3/2}\right)^{1/3}=x^{(3/2)\cdot(1/3)}=x^{1/2}$.
- So the whole innermost group $\sqrt[3]{x\sqrt{x}}$ is just $x^{1/2}$.
💡 Adding $1$ to the exponent then dividing it by $3$ sends $\tfrac{3}{2}$ back to $\tfrac{1}{2}$.
5.NF.B.4 Step 3 Notice the layer repeats
- The next layer out starts with that $x^{1/2}$ and does the same two operations: multiply by $x$ to get $x^{1}\cdot x^{1/2}=x^{3/2}$, then take a cube root to get $\left(x^{3/2}\right)^{1/3}=x^{1/2}$ again.
- Multiplying the exponent by $\tfrac{1}{3}$ is what resets $\tfrac{3}{2}$ back to $\tfrac{1}{2}$ every time.
- So each cube-root layer is a fixed point: no matter how deep, it keeps returning $x^{1/2}$.
💡 The product $\tfrac{3}{2}\times\tfrac{1}{3}=\tfrac{1}{2}$ is why every layer lands back on the same power.
8.EE.A.1 Step 4 Apply the outermost layer
- One layer is left.
- Multiply the running $x^{1/2}$ by $x$: $x^{1}\cdot x^{1/2}=x^{3/2}$.
- Take the final cube root: $\sqrt[3]{x^{3/2}}=x^{1/2}$.
- Written back as a root, $x^{1/2}=\sqrt{x}$.
- The entire tower simplifies to $\sqrt{x}$, which is choice (A).
💡 The outermost cube root does the same reset one last time, leaving $x^{1/2}=\sqrt{x}$.
8.EE.A.2 Turn each root into a fractional exponent so the operations become arithmetic on 8.EE.A.1 Start at the very center: $x\sqrt{x}=x^{1}\cdot x^{1/2}=x^{3/2}$ by adding expon 5.NF.B.4 The next layer out starts with that $x^{1/2}$ and does the same two operations: 8.EE.A.1 One layer is left. Multiply the running $x^{1/2}$ by $x$: $x^{1}\cdot x^{1/2}=x^ Review
Reasonableness: 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 $\tfrac{1}{54}$ or $\tfrac{80}{81}$ — that rules out (C), (D), and (E) on size alone. The exponent oscillates between $\tfrac{3}{2}$ and $\tfrac{1}{2}$, and because the tower ends on a cube root it lands on $\tfrac{1}{2}$, i.e. $\sqrt{x}$ — matching choice (A) and not (B)'s $x^{2/3}$.
Alternative: Guess and check with a perfect square. Take $x=64$: $\sqrt{64}=8$, so $x\sqrt{x}=64\cdot 8=512$ and $\sqrt[3]{512}=8$. The next layer gives $64\cdot 8=512$ and $\sqrt[3]{512}=8$ again, and the outer layer repeats to give $8$. The final value is $8=\sqrt{64}$, so the expression equals $\sqrt{x}$ — confirming (A). Trying $x=16$ gives $4=\sqrt{16}$ the same way.
CCSS standards used (min grade 8)
8.EE.A.2Use square root and cube root symbols to represent solutions and evaluate roots (Rewriting $\sqrt{x}$ and $\sqrt[3]{y}$ as the fractional powers $x^{1/2}$ and $y^{1/3}$, and reading $x^{1/2}$ back as $\sqrt{x}$.)8.EE.A.1Know and apply the properties of integer exponents to generate equivalent expressions (Using $x^{1}\cdot x^{1/2}=x^{3/2}$ (product rule) and $(x^{3/2})^{1/3}=x^{1/2}$ (power-of-a-power rule) to collapse each layer.)5.NF.B.4Multiply a fraction by a fraction (Computing the exponent product $\tfrac{3}{2}\times\tfrac{1}{3}=\tfrac{1}{2}$ that makes every cube-root layer reset to $x^{1/2}$.)
⭐ 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 $3$.
⭐ 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 $3$.
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