Rewriting Expressions with Radicals, Rational Exponents - Algebra 1

Simplify using exponent properties: $x^{1/2}\cdot x^{1/3}$ .
Answer: $x^{5/6}$ This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Once expressions are in exponential form with rational exponents, all the regular exponent properties work: to multiply $x^{1/2}$ · $x^{1/3}$ , we add the exponents just like with integer exponents: 1/2 + 1/3 = 3/6 + 2/6 = 5/6, giving $x^{5/6}$ . Start with $x^{1/2}$ · $x^{1/3}$ , and since the bases are the same, add the exponents: find a common denominator of 6, so 3/6 + 2/6 = 5/6, resulting in $x^{5/6}$ . Choice A is correct because it accurately adds the fractions 1/2 and 1/3 to get 5/6. A common error, like in choice B, might be multiplying the exponents instead of adding, giving 1/2 * 1/3 = 1/6, but remember, we add for multiplication of like bases. For simplifying with exponent properties, convert all radicals to exponential form first if they aren't already: √x becomes $x^{1/2}$ , ∛x becomes $x^{1/3}$ , etc. Then use your exponent rules (add when multiplying, subtract when dividing, multiply when doing power of a power). Finally, convert back to radical form if that's what's asked for!
The expression $27^{\frac{2}{3}} \cdot 9^{\frac{1}{2}}$ equals which of the following?
Answer: $27$ When you encounter expressions with fractional exponents, remember that these represent roots and powers combined. The key is to convert everything to the same base or simplify each term using the relationship $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$ . Let's work through this step by step. First, notice that both 27 and 9 are powers of 3: $27 = 3^3$ and $9 = 3^2$ . This allows us to rewrite the expression using the same base. For $27^{\frac{2}{3}}$ : Since $27 = 3^3$ , we have $(3^3)^{\frac{2}{3}} = 3^{3 \cdot \frac{2}{3}} = 3^2 = 9$ . For $9^{\frac{1}{2}}$ : Since $9 = 3^2$ , we have $(3^2)^{\frac{1}{2}} = 3^{2 \cdot \frac{1}{2}} = 3^1 = 3$ . Therefore: $27^{\frac{2}{3}} \cdot 9^{\frac{1}{2}} = 9 \cdot 3 = 27$ , which is choice C. Choice A (18) might result from calculating $27^{\frac{2}{3}} = 9$ correctly but then mistakenly finding $9^{\frac{1}{2}} = 2$ instead of 3. Choice B (81) could come from incorrectly computing $27^{\frac{2}{3}} = 27$ and $9^{\frac{1}{2}} = 3$ , giving $27 \cdot 3 = 81$ . Choice D (243) might result from adding the exponents incorrectly or making computational errors with the fractional powers. Study tip: Always look for common bases when dealing with exponential expressions. Converting to prime factorizations (like powers of 2, 3, 5) often makes fractional exponent problems much simpler.
Which of the following expressions is equivalent to $\sqrt{x} \cdot \sqrt[3]{x}$ when $x \geq 0$ ?
Answer: $x^{\frac{5}{6}}$ Convert to rational exponents: $\sqrt{x} \cdot \sqrt[3]{x} = x^{\frac{1}{2}} \cdot x^{\frac{1}{3}} = x^{\frac{1}{2} + \frac{1}{3}} = x^{\frac{3}{6} + \frac{2}{6}} = x^{\frac{5}{6}}$ . Choice B results from incorrectly adding $\frac{1}{2} + \frac{2}{3}$ instead of $\frac{1}{2} + \frac{1}{3}$ . Choice C represents only the cube root term. Choice D represents only the square root term multiplied incorrectly.
Rewrite $27^{2/3}$ using radical notation and evaluate.
Answer: $9$ This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Rational exponents give us an exponential way to write roots: $x^{1/n}$ means the nth root of x (like $x^{1/2}$ = √x and $x^{1/3}$ = ∛x), and more generally, $x^{m/n}$ means take the nth root of x, then raise it to the mth power (or do the power first, then the root—either order works!). For $27^{2/3}$ , we can interpret this as (∛27)² or as ∛(27²). Let's use the first approach: ∛27 = 3 (since 3³ = 27), then 3² = 9. We could verify with the second approach: 27² = 729, and ∛729 = 9. Choice A is correct because $27^{2/3}$ = (∛27)² = 3² = 9. Choice B (6) might come from incorrectly multiplying 3 × 2, while choice C (3) would be just the cube root without squaring, and choice D (18) might come from multiplying 27 × (2/3). For simplifying with exponent properties, convert all radicals to exponential form first if they aren't already: √x becomes $x^{1/2}$ , ∛x becomes $x^{1/3}$ , etc. Then use your exponent rules (add when multiplying, subtract when dividing, multiply when doing power of a power). Finally, convert back to radical form if that's what's asked for!
Rewrite $(\sqrt[4]{x})^3$ using rational exponents.
Answer: $x^{3/4}$ This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Rational exponents give us an exponential way to write roots: $x^{1/n}$ means the nth root of x (like $x^{1/2}$ = √x and $x^{1/3}$ = ∛x), and more generally, $x^{m/n}$ means take the nth root of x, then raise it to the mth power (or do the power first, then the root—either order works!). First, we convert ⁴√x to exponential form: ⁴√x = $x^{1/4}$ . Then we apply the power of 3: ( $x^{1/4}$ )³ = $x^{1/4 · 3}$ = $x^{3/4}$ . This uses the power rule: $(x^a)^b$ = $x^{ab}$ . Choice B is correct because (⁴√x)³ = ( $x^{1/4}$ )³ = $x^{3/4}$ . Choice A incorrectly inverts the fraction to get 4/3, while choice C multiplies the exponents incorrectly. For simplifying with exponent properties, convert all radicals to exponential form first if they aren't already: √x becomes $x^{1/2}$ , ∛x becomes $x^{1/3}$ , etc. Then use your exponent rules (add when multiplying, subtract when dividing, multiply when doing power of a power). Finally, convert back to radical form if that's what's asked for!
Rewrite $x^{2/3}$ using radicals.
Answer: $\sqrt[3]{x^2}$ This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Converting between radicals and rational exponents follows a simple pattern: the denominator of the exponent tells you which root (square root is 1/2, cube root is 1/3), and the numerator tells you what power. So ∛(x²) = $x^{2/3}$ because we're taking the cube root (denominator 3) of x squared (numerator 2). To convert $x^{2/3}$ to radical form, the denominator 3 tells us we need a cube root, and the numerator 2 tells us the power is 2, so $x^{2/3}$ = ∛(x²). This can also be written as (∛x)², which gives the same result. Choice B is correct because $x^{2/3}$ means the cube root (from denominator 3) of x squared (from numerator 2), which is written as ∛(x²). Choice A incorrectly gives √(x³), which would be $x^{3/2}$ , not $x^{2/3}$ - the numerator and denominator roles are switched. The key to converting: denominator of exponent = which root, numerator = which power. So $x^{3/4}$ means fourth root of x cubed: ⁴√(x³). To remember which is which, think '3 on top means power of 3, 4 on bottom means 4th root.' The fraction tells you everything!
Simplify using exponent properties: $x^{1/2}\cdot x^{1/3}$ .
Answer: $x^{5/6}$ This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Once expressions are in exponential form with rational exponents, all the regular exponent properties work: to multiply $x^{1/2}$ · $x^{1/3}$ , we add the exponents just like with integer exponents: 1/2 + 1/3 = 3/6 + 2/6 = 5/6, giving $x^{5/6}$ . When multiplying powers with the same base, we add the exponents: $x^{1/2}$ · $x^{1/3}$ = $x^{1/2 + 1/3}$ . To add these fractions, we need a common denominator: 1/2 = 3/6 and 1/3 = 2/6, so 1/2 + 1/3 = 3/6 + 2/6 = 5/6. Choice A is correct because $x^{1/2}$ · $x^{1/3}$ = $x^{1/2 + 1/3}$ = $x^{5/6}$ . Choice B incorrectly subtracts the exponents (1/2 - 1/3 = 1/6), which would be the rule for division, not multiplication. The key to converting: denominator of exponent = which root, numerator = which power. So $x^{3/4}$ means fourth root of x cubed: ⁴√(x³). To remember which is which, think '3 on top means power of 3, 4 on bottom means 4th root.' The fraction tells you everything!
Rewrite $\sqrt[3]{x^2}$ using rational exponent notation.
Answer: $x^{2/3}$ This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Converting between radicals and rational exponents follows a simple pattern: the denominator of the exponent tells you which root (square root is 1/2, cube root is 1/3), and the numerator tells you what power. So ∛(x²) = $x^{2/3}$ because we're taking the cube root (denominator 3) of x squared (numerator 2). For this problem, we have the cube root of x², which means the denominator is 3 (for cube root) and the numerator is 2 (for the power of x), giving us $x^{2/3}$ . Choice B is correct because it properly converts the cube root (denominator 3) of x squared (numerator 2) into the rational exponent 2/3. Choice A would mean the square root of x³, while choice C would mean the sixth root of x, neither matching our original expression. The key to converting: denominator of exponent = which root, numerator = which power. So $x^{3/4}$ means fourth root of x cubed: ⁴√(x³). To remember which is which, think '3 on top means power of 3, 4 on bottom means 4th root.' The fraction tells you everything!
Which of the following is equivalent to $\sqrt{x^3} \cdot \sqrt[4]{x}$ when $x \geq 0$ ?
Answer: $x^{\frac{7}{4}}$ When you see radical expressions multiplied together, you're working with exponent rules in disguise. The key insight is converting radicals to fractional exponents, then using the rule that when multiplying powers with the same base, you add the exponents. First, convert each radical to exponential form. Remember that $\sqrt{x^3} = (x^3)^{\frac{1}{2}}$ and $\sqrt[4]{x} = x^{\frac{1}{4}}$ . Using the power rule $(a^m)^n = a^{mn}$ , we get $\sqrt{x^3} = x^{3 \cdot \frac{1}{2}} = x^{\frac{3}{2}}$ . Now multiply: $x^{\frac{3}{2}} \cdot x^{\frac{1}{4}}$ . When multiplying powers with the same base, add exponents: $\frac{3}{2} + \frac{1}{4}$ . To add these fractions, find a common denominator: $\frac{3}{2} = \frac{6}{4}$ , so $\frac{6}{4} + \frac{1}{4} = \frac{7}{4}$ . Therefore, the answer is $x^{\frac{7}{4}}$ , which is choice B. Looking at the wrong answers: Choice A ( $x^{\frac{5}{4}}$ ) results from incorrectly adding $1 + \frac{1}{4}$ instead of $\frac{3}{2} + \frac{1}{4}$ . Choice C ( $x^{\frac{3}{2}}$ ) is just the first term, suggesting you forgot to include the second radical entirely. Choice D ( $x^{\frac{13}{4}}$ ) might come from multiplying the exponents instead of adding them. Remember: when you see radicals being multiplied, immediately convert to fractional exponents. This transforms a potentially confusing radical problem into straightforward exponent arithmetic that follows familiar rules.
Which expression is equivalent to $\frac{\sqrt[3]{x^4}}{\sqrt{x}}$ when $x > 0$ ?
Answer: $x^{\frac{5}{6}}$ When you encounter expressions with radicals and need to simplify them, converting everything to exponential form using fractional exponents makes the algebra much cleaner. First, rewrite each radical using fractional exponents. Remember that $\sqrt[n]{x^m} = x^{\frac{m}{n}}$ , so $\sqrt[3]{x^4} = x^{\frac{4}{3}}$ and $\sqrt{x} = x^{\frac{1}{2}}$ . This transforms our expression into: $\frac{x^{\frac{4}{3}}}{x^{\frac{1}{2}}}$ Now apply the quotient rule for exponents: when dividing powers with the same base, subtract the exponents. So we get: $x^{\frac{4}{3} - \frac{1}{2}}$ To subtract these fractions, find a common denominator. The LCD of 3 and 2 is 6: $\frac{4}{3} = \frac{8}{6}$ and $\frac{1}{2} = \frac{3}{6}$ Therefore: $x^{\frac{8}{6} - \frac{3}{6}} = x^{\frac{5}{6}}$ Answer choice A ( $x^{\frac{3}{2}}$ ) likely comes from incorrectly adding the exponents instead of subtracting. Answer choice B ( $x^{\frac{7}{6}}$ ) results from adding $\frac{4}{3} + \frac{1}{2}$ when you should subtract. Answer choice D ( $x^{\frac{2}{3}}$ ) might come from misapplying the radical conversion rules or arithmetic errors with the fractions. Study tip: Always convert radicals to fractional exponents when simplifying complex expressions. This lets you use familiar exponent rules instead of wrestling with radical notation. Practice fraction arithmetic with different denominators—it's crucial for these problems.
Which is equivalent to $\sqrt{x^3}$ ? (Assume $x\ge 0$ .)
Answer: $x^{3/2}$ This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Converting between radicals and rational exponents follows a simple pattern: the denominator of the exponent tells you which root (square root is 1/2, cube root is 1/3), and the numerator tells you what power. So ∛(x²) = $x^{2/3}$ because we're taking the cube root (denominator 3) of x squared (numerator 2). For √(x³), we have a square root (denominator 2) of x cubed (numerator 3), which gives us $x^{3/2}$ . The square root means the denominator is 2, and x³ means the numerator is 3, resulting in the rational exponent 3/2. Choice A is correct because √(x³) = $x^{3/2}$ , with the square root giving denominator 2 and the cube giving numerator 3. Choice B ( $x^{2/3}$ ) would be ∛(x²), choice C ( $x^{1/3}$ ) would be just ∛x, and choice D ( $x^{3/1}$ = x³) would be x cubed without any root. The key to converting: denominator of exponent = which root, numerator = which power. So $x^{3/4}$ means fourth root of x cubed: ⁴√(x³). To remember which is which, think '3 on top means power of 3, 4 on bottom means 4th root.' The fraction tells you everything!
Which expression is equivalent to $\sqrt{x^3}$ ? (Write in rational exponent form.)
Answer: $x^{3/2}$ This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Converting between radicals and rational exponents follows a simple pattern: the denominator of the exponent tells you which root (square root is 1/2, cube root is 1/3), and the numerator tells you what power. So ∛(x²) = $x^{2/3}$ because we're taking the cube root (denominator 3) of x squared (numerator 2). To convert √(x³), we identify that the square root sign (no little number) means the denominator is 2, and the power inside is 3, which becomes the numerator, giving us $x^{3/2}$ . This means we're taking the square root of x cubed. Choice B is correct because √(x³) properly converts to $x^{3/2}$ , where the square root gives denominator 2 and the power 3 inside gives numerator 3. Choice A incorrectly gives $x^{2/3}$ , which would represent the cube root of x squared, not the square root of x cubed. The key to converting: denominator of exponent = which root, numerator = which power. So $x^{3/4}$ means fourth root of x cubed: ⁴√(x³). To remember which is which, think '3 on top means power of 3, 4 on bottom means 4th root.' The fraction tells you everything!
What is $\sqrt[3]{64}$ written in rational exponent form?
Answer: $64^{1/3}$ This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Converting between radicals and rational exponents follows a simple pattern: the denominator of the exponent tells you which root (square root is 1/2, cube root is 1/3), and the numerator tells you what power. So ∛(x²) = $x^{2/3}$ because we're taking the cube root (denominator 3) of x squared (numerator 2). To convert ∛64 to rational exponent form, we recognize that the cube root sign means the denominator is 3, and since there's no visible power inside (just 64), the numerator is 1, giving us $64^{1/3}$ . This represents taking the cube root of 64. Choice C is correct because ∛64 = $64^{1/3}$ , where the cube root gives us the denominator 3 and the implicit power of 1 gives us the numerator. Choice A incorrectly gives $64^{1/2}$ , which would mean √64 (square root), not ∛64 (cube root). The key to converting: denominator of exponent = which root, numerator = which power. So $x^{3/4}$ means fourth root of x cubed: ⁴√(x³). To remember which is which, think '3 on top means power of 3, 4 on bottom means 4th root.' The fraction tells you everything!
Rewrite the rational exponent expression $x^{3/4}$ using radicals.
Answer: $\sqrt[4]{x^3}$ This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Converting between radicals and rational exponents follows a simple pattern: the denominator of the exponent tells you which root (square root is 1/2, cube root is 1/3), and the numerator tells you what power. So ∛(x²) = $x^{2/3}$ because we're taking the cube root (denominator 3) of x squared (numerator 2). For $x^{3/4}$ , the denominator 4 tells us we need a fourth root, and the numerator 3 tells us the power is 3. This gives us ⁴√(x³), which means the fourth root of x cubed. Choice C is correct because $x^{3/4}$ means the fourth root (denominator 4) of x cubed (numerator 3), written as ⁴√(x³). Choice A incorrectly uses a cube root instead of a fourth root, while choice B uses a square root with the wrong power. The key to converting: denominator of exponent = which root, numerator = which power. So $x^{3/4}$ means fourth root of x cubed: ⁴√(x³). To remember which is which, think '3 on top means power of 3, 4 on bottom means 4th root.' The fraction tells you everything!
Rewrite $27^{2/3}$ using radical notation and evaluate.
Answer: $9$ This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Rational exponents give us an exponential way to write roots: $x^{1/n}$ means the nth root of x (like $x^{1/2}$ = √x and $x^{1/3}$ = ∛x), and more generally, $x^{m/n}$ means take the nth root of x, then raise it to the mth power (or do the power first, then the root—either order works!). To evaluate $27^{2/3}$ , we can first rewrite it as (∛27)² or as ∛(27²). Using the first approach: ∛27 = 3 (since 3³ = 27), then 3² = 9. Using the second approach: 27² = 729, then ∛729 = 9 (since 9³ = 729). Choice C is correct because $27^{2/3}$ = (∛27)² = 3² = 9. Choice A gives just the cube root without squaring it, while choices B and D represent different incorrect calculations. For simplifying with exponent properties, convert all radicals to exponential form first if they aren't already: √x becomes $x^{1/2}$ , ∛x becomes $x^{1/3}$ , etc. Then use your exponent rules (add when multiplying, subtract when dividing, multiply when doing power of a power). Finally, convert back to radical form if that's what's asked for!
If $\sqrt[3]{x^2} = x^{\frac{a}{b}}$ where $\frac{a}{b}$ is in lowest terms, what is the value of $a + b$ ?
Answer: $5$ When you encounter expressions with radicals and exponents, the key is converting everything to exponential form using the rule that $\sqrt[n]{x^m} = x^{\frac{m}{n}}$ . Let's rewrite the left side of the equation. The cube root $\sqrt[3]{x^2}$ can be expressed as $x^{\frac{2}{3}}$ . This means our equation becomes: $x^{\frac{2}{3}} = x^{\frac{a}{b}}$ Since the bases are the same (both are $x$ ), the exponents must be equal: $\frac{a}{b} = \frac{2}{3}$ We need to check if $\frac{2}{3}$ is already in lowest terms. Since 2 and 3 share no common factors other than 1, this fraction is already reduced. Therefore, $a = 2$ and $b = 3$ , giving us $a + b = 2 + 3 = 5$ . Looking at the wrong answers: Choice A (8) might come from incorrectly adding exponents or misapplying power rules. Choice B (3) represents just the value of $b$ alone, which suggests forgetting to add $a$ . Choice C (6) could result from multiplying $a \times b = 2 \times 3 = 6$ instead of adding them. The answer is D. Study tip: Always convert radicals to fractional exponents first—use $\sqrt[n]{x^m} = x^{\frac{m}{n}}$ . Then remember that when bases are equal, exponents must be equal. Finally, double-check that your fraction is in lowest terms by ensuring the numerator and denominator have no common factors.
Evaluate $27^{2/3}$ .
Answer: $9$ This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Rational exponents give us an exponential way to write roots: $x^{1/n}$ means the nth root of x (like $x^{1/2}$ = √x and $x^{1/3}$ = ∛x), and more generally, $x^{m/n}$ means take the nth root of x, then raise it to the mth power (or do the power first, then the root—either order works!). To evaluate $27^{2/3}$ , we can think of it as (∛27)² or ∛(27²). Since ∛27 = 3 (because 3³ = 27), we get 3² = 9. Alternatively, 27² = 729, and ∛729 = 9. Choice C is correct because $27^{2/3}$ = (∛27)² = 3² = 9. Choice A gives 6, which might come from incorrectly calculating 27 × 2/3 = 18, then dividing by 3, but that's not how rational exponents work. For simplifying with exponent properties, convert all radicals to exponential form first if they aren't already: √x becomes $x^{1/2}$ , ∛x becomes $x^{1/3}$ , etc. Then use your exponent rules (add when multiplying, subtract when dividing, multiply when doing power of a power). Finally, convert back to radical form if that's what's asked for!
Simplify using exponent properties: $\dfrac{x^{5/6}}{x^{1/2}}$ .
Answer: $x^{1/3}$ This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Once expressions are in exponential form with rational exponents, all the regular exponent properties work: to multiply $x^{1/2}$ · $x^{1/3}$ , we add the exponents just like with integer exponents: 1/2 + 1/3 = 3/6 + 2/6 = 5/6, giving $x^{5/6}$ . When dividing powers with the same base, we subtract the exponents: $x^{5/6}$ ÷ $x^{1/2}$ = $x^{5/6 - 1/2}$ . To subtract these fractions, find a common denominator: 5/6 - 1/2 = 5/6 - 3/6 = 2/6 = 1/3. Choice B is correct because $x^{5/6}$ ÷ $x^{1/2}$ = $x^{5/6 - 3/6}$ = $x^{2/6}$ = $x^{1/3}$ . Choice A incorrectly gives $x^{5/12}$ , which might come from multiplying the denominators instead of finding a common denominator. For simplifying with exponent properties, convert all radicals to exponential form first if they aren't already: √x becomes $x^{1/2}$ , ∛x becomes $x^{1/3}$ , etc. Then use your exponent rules (add when multiplying, subtract when dividing, multiply when doing power of a power). Finally, convert back to radical form if that's what's asked for!
Rewrite $\sqrt[3]{x^5}$ using rational exponents.
Answer: $x^{5/3}$ This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Converting between radicals and rational exponents follows a simple pattern: the denominator of the exponent tells you which root (square root is 1/2, cube root is 1/3), and the numerator tells you what power. So ∛(x²) = $x^{2/3}$ because we're taking the cube root (denominator 3) of x squared (numerator 2). For ∛( $x^5$ ), the cube root means exponent 1/3, and the power 5 inside means multiply by 5, so $x^{5/3}$ . Choice A is correct because it places 5 in the numerator for the power and 3 in the denominator for the root. A mistake like in choice B could be inverting the fraction, giving $x^{3/5}$ which would be the fifth root of x cubed instead. The key to converting: denominator of exponent = which root, numerator = which power. So $x^{3/4}$ means fourth root of x cubed: ∜(x³). To remember which is which, think '3 on top means power of 3, 4 on bottom means 4th root.' The fraction tells you everything!
Rewrite the radical expression $\sqrt[3]{x^2}$ using rational exponents.
Answer: $x^{2/3}$ This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Converting between radicals and rational exponents follows a simple pattern: the denominator of the exponent tells you which root (square root is 1/2, cube root is 1/3), and the numerator tells you what power. So ∛(x²) = $x^{2/3}$ because we're taking the cube root (denominator 3) of x squared (numerator 2). To convert the radical ∛(x²), recognize that the cube root means exponent 1/3, and the power of 2 inside means multiply by 2, so overall $x^{2/3}$ . Choice B is correct because it properly applies the conversion: the index 3 becomes the denominator, and the exponent 2 becomes the numerator, giving $x^{2/3}$ . A common mistake, like in choice A, is swapping the numerator and denominator, which would incorrectly give $x^{3/2}$ for a square root of x cubed instead. The key to converting: denominator of exponent = which root, numerator = which power. So $x^{3/4}$ means fourth root of x cubed: ∜(x³). To remember which is which, think '3 on top means power of 3, 4 on bottom means 4th root.' The fraction tells you everything!

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