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!

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!

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 ∛(x²), we have a cube root (which means the denominator is 3) of x squared (which means the numerator is 2), giving us x^(2/3). Choice B is correct because it properly converts the cube root to a denominator of 3 and the power of 2 to the numerator. Choice A incorrectly uses 3/2, which would represent √(x³), not ∛(x²). 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!

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!

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 $x^{1/2} = \sqrt{x}$ and $x^{1/3} = \sqrt[3]{x}$), 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 $16^{3/4}$, first convert to radical form: $16^{3/4} = \sqrt[4]{16^3}$ or $(\sqrt[4]{16})^3$. Since $\sqrt[4]{16} = 2$ (because $2^4 = 16$), we get $2^3 = 8$. Choice C is correct because $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$. Choice B gives 12, which might come from incorrectly calculating 16 × 3/4 = 12, 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: $\sqrt{x}$ becomes $x^{1/2}$, $\sqrt[3]{x}$ 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!

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 ⁵√(x³), we have the fifth root (denominator 5) of x cubed (numerator 3). This converts directly to x^(3/5). We can verify this: the fifth root is represented by the exponent 1/5, and x³ raised to the 1/5 power gives us x^(3·1/5) = x^(3/5). Choice B is correct because ⁵√(x³) means the fifth root (denominator 5) of x cubed (numerator 3), giving us x^(3/5). Choice A incorrectly inverts the fraction to 5/3, which would represent the cube root of x⁵, not the fifth root of x³. 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!

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!

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!

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!

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 16^(3/4), we can rewrite it as (⁴√16)³ or as ⁴√(16³). Using the first approach: ⁴√16 = 2 (since 2⁴ = 16), then 2³ = 8. We can verify: 16³ = 4096, and ⁴√4096 = 8 (since 8⁴ = 4096). Choice B is correct because 16^(3/4) = (⁴√16)³ = 2³ = 8. Choice A gives just the fourth root without cubing it, while choices C 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!