📝 Power functions with rational exponents (13 MCQs)

📖 From Calculus • 1. Basics before calculus • 13 questions available

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What is Power functions with rational exponents?

Definition:
Power functions with rational exponents are of the form $y = x^{m/n}$ , where $m/n$ is a fraction in lowest terms, which can be interpreted as $y = \sqrt[n]{x^m}$ or $(\sqrt[n]{x})^m$ , and their domains depend on whether $n$ is odd or even.

Example:
For $y = x^{1/2} = \sqrt{x}$ , domain is $x \geq 0$ ; for $y = x^{2/3}$ , domain is all real numbers since $\sqrt[3]{x^2}$ is defined for all $x$ .

Reason:
These functions bridge roots and powers, allowing us to model growth like population (with fractional exponents) or geometric scaling.

Easy

Medium

Hard

📝 All Power functions with rational exponents MCQs

Q1. Which radical notation correctly represents the function $f(x)=x^{1/n}$ for a positive integer $n$ ?

\sqrt[n]{x}

✅

{n}\sqrt{x}

\sqrt{x^{n}}

x^{n}

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The notation $\sqrt[n]{x}$ is defined as the nth root of x, which is exactly the same as raising x to the power $1/n$ . The other choices either misplace the index or represent different powers, so option A correctly matches the given function.

Q2. Given $f(x)=\sqrt[4]{x}$ and $g(x)=x^{2}$ , which statement is true about the composition values at $x=16$ ?

f(g(16)) > g(f(16))

f(g(16)) = g(f(16))

✅

f(g(16)) < g(f(16))

D.Both are undefined

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Compute $g(16)=256$ ; then $f(g(16))=256^{1/4}=4$ . Also $f(16)=16^{1/4}=2$ and $g(f(16))=2^{2}=4$ . Both compositions give the same result, so the equality statement is correct.

Q3. How does the graph of $y=\sqrt{x}$ compare to the graph of $y=x^{1/3}$ for $x\ge 0$ ?

A.Both are linear

\sqrt{x}

grows faster than

x^{1/3}

✅

x^{1/3}

grows faster than

\sqrt{x}

D.They coincide for all

x\ge0

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: For $x>1$ , the exponent 0.5 (square root) yields larger values than the exponent 0.333… (cube root). Near zero both approach zero, but the square‑root curve rises more quickly, making option B the accurate comparison.

Q4. If a power function has exponent $p=1/n$ with $n$ even, which must be true about its domain?

A.All real numbers

B.Only non‑negative numbers ✅

C.Only negative numbers

D.No real numbers

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: When $n$ is even, the nth root of a negative number is not real, so the function is defined only for $x\ge0$ . Hence the domain consists of non‑negative real numbers, matching option B.

Q5. Suppose $h(x)=x^{1/5}$ . Which statement correctly describes the effect of squaring the input before applying $h$ versus applying $h$ before squaring, for any positive $x$ ?

h(x^{2}) = [h(x)]^{2}

✅

h(x^{2}) > [h(x)]^{2}

h(x^{2}) < [h(x)]^{2}

D.Equality holds only when

x=1

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: $h(x^{2})=(x^{2})^{1/5}=x^{2/5}$ . Meanwhile $[h(x)]^{2}=(x^{1/5})^{2}=x^{2/5}$ . The two expressions are identical for every positive $x$ , confirming the equality in option A.

Q6. Consider the functions $f(x)=x^{1/2}$ and $g(x)=x^{2/3}$ . For $x>1$ , which function yields larger output values?

f(x)

for all

x>1

g(x)

for all

x>1

✅

C.They are equal

D.Depends on whether

x

is an integer

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The exponent of $g$ is $2/3\approx0.667$ , larger than the exponent $1/2=0.5$ of $f$ . For numbers greater than one, a larger exponent produces a larger value, so $g(x)$ is greater for every $x>1$ .}

Q7. When the exponent $p$ in $y=x^{p}$ is a reciprocal of an odd integer, how does the function behave for negative inputs?

A.Undefined

B.Produces positive outputs only

C.Mirrors the positive‑x graph

D.Produces negative outputs mirroring the positive‑x graph ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: If the denominator is odd, the root of a negative number is defined and remains negative, giving the function odd symmetry. Thus the graph for negative $x$ is the mirror image of the positive side, reflected across the origin, as described in option D.

Q8. Let $k(x)=x^{1/6}$ . Which inequality correctly represents the relationship between $k(64)$ and $k(8)^{2}$ ?

k(64) > k(8)^{2}

k(64) = k(8)^{2}

✅

k(64) < k(8)^{2}

D.No comparison possible because one side is undefined

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: $k(64)=64^{1/6}=2$ . Meanwhile $k(8)=8^{1/6}=2^{1/2}=\sqrt{2}$ ; squaring gives $(\sqrt{2})^{2}=2$ . Both sides equal 2, making the equality statement correct.

Q9. Which statement is true about the derivative of $f(x)=x^{1/n}$ for a positive integer $n$ ?

f&#039;(x)=\frac{1}{n}x^{1/n-1}

✅

f&#039;(x)=\frac{1}{n}x^{1/n+1}

f&#039;(x)=n x^{n-1}

D.The derivative does not exist for any

x

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Using the power rule, $\frac{d}{dx}x^{p}=p\,x^{p-1}$ . Substituting $p=1/n$ gives $f'(x)=\frac{1}{n}x^{1/n-1}$ , which matches option A. The other choices misuse the rule or claim non‑existence incorrectly.

Q10. If a function $y=x^{p}$ with $p=1/4$ is reflected over the line $y=x$ , which function describes the reflected graph?

y=x^{4}

✅

y=\sqrt[4]{x}

y=x^{1/2}

y=x^{2}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Reflecting over $y=x$ swaps the variables, turning $y=x^{1/4}$ into $x=y^{1/4}$ . Solving for $y$ yields $y=x^{4}$ . Hence the reflected graph is represented by $y=x^{4}$ , which is option A.

Q11. For $f(x)=x^{1/3}$ and $g(x)=x^{2}$ , determine which composition yields a larger value at $x=0.001$ .

f(g(x))

is larger

g(f(x))

is larger

C.They are equal ✅

D.Both are zero

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Compute $g(0.001)=10^{-6}$ ; then $f(g)= (10^{-6})^{1/3}=10^{-2}=0.01$ . Also $f(0.001)=10^{-1}=0.1$ ; $g(f)=0.1^{2}=0.01$ . Both compositions give 0.01, so the values are equal, confirming option C.

Q12. Consider $f(x)=x^{1/5}$ and $h(x)=x^{3/5}$ . Which correctly describes the ratio $\frac{h(x)}{f(x)}$ for $x>0$ ?

A.It equals

x^{2/5}

✅

B.It equals

x^{1/5}

C.It equals

x^{3/5}

D.It is constant equal to 1

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Dividing the two power functions gives $x^{3/5}/x^{1/5}=x^{(3/5-1/5)}=x^{2/5}$ . Therefore the ratio simplifies to $x^{2/5}$ , matching option A.

Q13. A student claims that the graph of $y=x^{1/3}$ is a horizontal stretch of the graph of $y=x^{1/2}$ . Which reasoning correctly evaluates this claim?

A.True, because both are root functions

B.False; the two graphs differ in curvature and cannot be obtained by a simple stretch ✅

C.True; a stretch factor of

x^{1/6}

converts one into the other

D.False; only a vertical stretch could relate them

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Although both are root functions, their exponents (1/3 vs 1/2) give distinct curvature. No constant horizontal stretch can transform one graph into the other; the shapes are fundamentally different, so the claim is false as stated in option B.

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