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Algebra Help: Rewriting Expressions With Radicals Rational Exponents

Review real example questions for Rewriting Expressions With Radicals Rational Exponents in Algebra.

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If $\sqrt[3]{x^2} = x^{\frac{a}{b}}$ where $\frac{a}{b}$ is in lowest terms, what is the value of $a + b$ ?

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Question 1

If $\sqrt[3]{x^2} = x^{\frac{a}{b}}$ where $\frac{a}{b}$ is in lowest terms, what is the value of $a + b$ ?

$8$
$3$
$6$
$5$ (correct answer)

Explanation: 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.

Question 2

Express $\sqrt{x}$ using a rational exponent.

$x^{2}$
$x^{-1/2}$
$x^{1/3}$
$x^{1/2}$ (correct answer)

Explanation: 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!). The square root symbol √x means the second root of x, which in exponential form is x^(1/2). The denominator 2 indicates it's a square root, and the numerator 1 indicates we're taking x to the first power. Choice A is correct because √x = x^(1/2), following the pattern where the root index becomes the denominator of the exponent. Choice C would represent ∛x (cube root), not √x (square 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!

Question 3

Rewrite $(\sqrt[4]{x})^3$ using rational exponents.

$x^{4/3}$
$x^{3/4}$ (correct answer)
$x^{1/12}$
$x^{7/4}$

Explanation: 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, convert ⁴√x to exponential form: ⁴√x = x^(1/4). Then apply the power of 3: (x^(1/4))³ = x^(1/4 × 3) = x^(3/4). Choice B is correct because (⁴√x)³ = (x^(1/4))³ = x^(3/4), properly applying the power rule for exponents. Choice A would be x^(4/3), which reverses the role of the root and 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!

Question 4

Evaluate $27^{2/3}$ .

$6$
$12$
$9$ (correct answer)
$18$

Explanation: 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!

Question 5

Rewrite $16^{3/4}$ using radicals, then evaluate.

$12$
$8$ (correct answer)
$4$
$6$

Explanation: 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!

Question 6

Rewrite $27^{2/3}$ using radical notation and evaluate.

$3$
$6$
$9$ (correct answer)
$18$

Explanation: 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!

Question 7

What is the value of $16^{3/4}$ ?

$8$ (correct answer)
$12$
$4$
$6$

Explanation: 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 16^{3/4}, you can compute it as (16^{1/4})^3: the fourth root of 16 is 2 (since $2^4$ =16), and 2^3=8. Alternatively, (16^3)^{1/4} = 4096^{1/4}, and since 8^4=4096, the fourth root is 8. Choice B is correct because both approaches yield 8. A distractor like choice D might stop at the fourth root, getting 2, but that forgets to apply the numerator 3 as a power. 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!

Question 8

Rewrite $27^{2/3}$ using radical notation and evaluate.

$6$
$9$ (correct answer)
$18$
$3$

Explanation: 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!

Question 9

Evaluate $16^{3/4}$ .

$4$
$8$ (correct answer)
$12$
$16$

Explanation: 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!

Question 10

Rewrite $27^{2/3}$ using radical notation and evaluate.

$3$
$6$
$9$ (correct answer)
$18$

Explanation: 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}, you can rewrite it as (27^{1/3})^2: the cube root of 27 is 3, and 3 squared is 9. Alternatively, (27^2)^{1/3} = 729^{1/3}, and since 9^3 = 729, the cube root is 9. Choice C is correct because both methods confirm the value is 9, matching the evaluation after converting to radical form like ∛( $27^2$ ). A distractor like choice A might come from just taking the cube root without the power, giving 3, but that ignores the numerator 2. 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!