AMC 10 · 2008 · #3
Grade 8 algebraAssume that x is a positive real number. Which is equivalent to 3xx?
Pick an answer.
AMC 10 2008 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: Rewrite the expression $\sqrt[3]{x\sqrt{x}}$, where $x$ is a positive real number, as a single power of $x$.
Givens: $x$ is a positive real number; The expression $\sqrt[3]{x\sqrt{x}}$ — a cube root wrapping the product of $x$ and $\sqrt{x}$; Answer choices: (A) $x^{1/6}$, (B) $x^{1/4}$, (C) $x^{3/8}$, (D) $x^{1/2}$, (E) $x$
Unknowns: The single exponent $r$ for which $\sqrt[3]{x\sqrt{x}}=x^{r}$
Understand
Restated: Rewrite the expression $\sqrt[3]{x\sqrt{x}}$, where $x$ is a positive real number, as a single power of $x$.
Givens: $x$ is a positive real number; The expression $\sqrt[3]{x\sqrt{x}}$ — a cube root wrapping the product of $x$ and $\sqrt{x}$; Answer choices: (A) $x^{1/6}$, (B) $x^{1/4}$, (C) $x^{3/8}$, (D) $x^{1/2}$, (E) $x$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #6 Guess and Check, #3 Eliminate Possibilities
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.
Execute — Answer: D
8.EE.A.1 Step 1 Turn radicals into fractional exponents
- A square root is the $\tfrac12$ power and a cube root is the $\tfrac13$ power.
- So $\sqrt{x}=x^{1/2}$, and $\sqrt[3]{\ }$ means raising to the $\tfrac13$ power.
- The expression becomes $\left(x\cdot x^{1/2}\right)^{1/3}$, with the base $x$ appearing everywhere.
💡 Every root is just a fractional power, so rewriting the roots turns the problem into plain exponent arithmetic.
5.NF.A.1 Step 2 Combine the product inside
- Inside the cube root, $x$ means $x^{1}$, and multiplying powers of the same base adds their exponents: $x^{1}\cdot x^{1/2}=x^{1+1/2}$.
- Adding the exponents, $1+\tfrac12=\tfrac32$, so the inside becomes $x^{3/2}$.
💡 Same base times same base means you just add the little exponents, and $1+\tfrac12$ is three-halves.
5.NF.B.4 Step 3 Apply the outer cube root
- Now take the cube root, i.e.
- raise $x^{3/2}$ to the $\tfrac13$ power.
- A power of a power multiplies the exponents: $\left(x^{3/2}\right)^{1/3}=x^{(3/2)\cdot(1/3)}$.
- Multiplying the fractions, $\tfrac32\cdot\tfrac13=\tfrac{3}{6}=\tfrac12$, so the expression equals $x^{1/2}$.
💡 Stacking a power onto a power multiplies the exponents, and three-halves times one-third is one-half.
8.EE.A.2 Step 4 Check by plugging in $x=64$
- Test with a friendly number.
- Let $x=64$.
- Then $\sqrt{64}=8$, so $x\sqrt{x}=64\cdot8=512$, and $\sqrt[3]{512}=8$.
- The winning form must also give $8$: $x^{1/2}=\sqrt{64}=8$, a match.
- The decoys miss — $x^{1/6}=2$ (what you get by wrongly multiplying all of $1,\tfrac12,\tfrac13$), $x^{1/4}=\sqrt[4]{64}\approx2.83$, $x^{3/8}\approx4.76$, and $x=64$ (skipping the roots entirely).
- Only $x^{1/2}$ lands on $8$, so the answer is (D).
💡 One well-chosen number turns the abstract expression into a plain arithmetic check that only the right answer can pass.
8.EE.A.1 A square root is the $\tfrac12$ power and a cube root is the $\tfrac13$ power. S 5.NF.A.1 Inside the cube root, $x$ means $x^{1}$, and multiplying powers of the same base 5.NF.B.4 Now take the cube root, i.e. raise $x^{3/2}$ to the $\tfrac13$ power. A power of 8.EE.A.2 Test with a friendly number. Let $x=64$. Then $\sqrt{64}=8$, so $x\sqrt{x}=64\cd Review
Reasonableness: Each root shrinks the exponent: the product inside builds up to $x^{3/2}$, then the cube root cuts that exponent to a third, landing at $x^{1/2}$ — a sensible middle value between $x^{0}=1$ and $x^{1}=x$. The numeric test seals it: at $x=64$ the expression is $8$, and $64^{1/2}=8$ while every other choice misses. Choice (A) $x^{1/6}$ is the trap for multiplying the three exponents $1\cdot\tfrac12\cdot\tfrac13$ instead of adding then multiplying, and (E) $x$ is what you get by dropping the roots altogether.
Alternative: Clear the fractions with a common index instead. Since $x\sqrt{x}=\sqrt{x^{2}\cdot x}=\sqrt{x^{3}}$, the whole expression is $\sqrt[3]{\sqrt{x^{3}}}=\sqrt[6]{x^{3}}=x^{3/6}=x^{1/2}$. A cube root outside a square root is a sixth root, and $x^{3}$ under a sixth root is $x^{1/2}$ — the same (D).
CCSS standards used (min grade 8)
8.EE.A.1Know and apply the properties of integer exponents (Rewriting the radicals as fractional exponents and using the same-base product and power-of-a-power rules.)5.NF.A.1Add and subtract fractions with unlike denominators (Adding the exponents $1+\tfrac12=\tfrac32$ when combining $x\cdot\sqrt{x}$.)5.NF.B.4Apply and extend understanding of multiplication to multiply a fraction by a fraction (Multiplying the exponents $\tfrac32\cdot\tfrac13=\tfrac12$ when applying the cube root.)8.EE.A.2Use square root and cube root symbols to represent solutions (Evaluating $\sqrt{64}=8$ and $\sqrt[3]{512}=8$ in the numerical check.)
⭐ 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 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.
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