For what value of x is the given expression equivalent to (70n)^{30x}, where n > 1? \sqrt[5]{70n}\left(\sqrt{70n}\right)^2

C. \frac{1}{25}. The correct answer is $\frac{1}{25}$. An expression of the form $\sqrt[k]{a}$, where $k$ is an integer greater than 1 and $a \geq 0$, is equivalent to $a^{\frac{1}{k}}$. Therefore, the given expression, where $n > 1$, is equivalent to $(70n)^{\frac{1}{5}} \left((70n)^{\frac{1}{2}}\right)^2$. Applying properties of exponents, this expression can be rewritten as $(70n)^{\frac{1}{5}} (70n)^{\frac{1}{2} \cdot 2}$, or $(70n)^{\frac{1}{5}} (70n)^{1}$, which can be rewritten as $(70n)^{\frac{1}{5} + 1}$, or $(70n)^{\frac{6}{5}}$. It's given that the expression $\sqrt[5]{70n}\left(\sqrt{70n}\right)^2$ is equivalent to $(70n)^{30x}$, where $n > 1$. It follows that $(70n)^{\frac{6}{5}}$ is equivalent to $(70n)^{30x}$. Therefore, $\frac{6}{5} = 30x$. Dividing both sides of this equation by 30 yields $x = \frac{6}{150}$, which simplifies to $\frac{1}{25}$. Thus, the value of $x$ is $\frac{1}{25}$.

If \frac{\sqrt{x^5}}{\sqrt[3]{x^4}} = x^{\frac{a}{b}} for all positive values of x, what is the value of \frac{a}{b}?

A. \frac{7}{6}. Choice A is correct. Rewrite the radicals using rational exponents: $\sqrt{x^5}=x^{5/2}$ and $\sqrt[3]{x^4}=x^{4/3}$. Then use the quotient rule for exponents: $x^{5/2}/x^{4/3}=x^{5/2-4/3}$. Subtracting gives $\frac{15}{6}-\frac{8}{6}=\frac{7}{6}$, so $x^{5/2-4/3}=x^{7/6}$. Therefore, $\frac{a}{b}=\frac{7}{6}$, which is choice A. Choice B is incorrect because it comes from subtracting the exponents incorrectly. Choice C is incorrect because it comes from adding instead of subtracting. Choice D is incorrect because it does not match the simplified exponent.

SAT Math skill page

SAT Exponent Rules Practice

Use exponent rules to simplify expressions, compare powers, and avoid common sign traps.

Practice exponent rules in Dolphin SAT Practice questions

10-16 min practice time3 questions on pageAdvanced Math

Practice time10-16 min

Question bank3 questions

Best forAdvanced Math

What this tests

What to know for this SAT skill

Apply product and quotient rules for exponents
Simplify powers raised to powers
Interpret negative and zero exponents
Connect radicals to fractional exponents

Dolphin question bank

Practice your skill

Question 1 of 3Medium

Which expression represents the product of (x^{-6}

y^{3} z^{5}

) and (

x^{4} z^{5}

y^{8} z^{- 7}

$x^{- 2} z^{10} + y^{11} z^{- 2}$
$x^{- 2} z^{10} + x^{- 6} z^{- 2}$
$x^{- 2} y^{3} z^{10} + y^{8} z^{- 7}$
$x^{- 2} y^{3} z^{10} + x^{- 6} y^{11} z^{- 2}$

Show answer and solution

Correct answer $D . x^{- 2} y^{3} z^{10} + x^{- 6} y^{11} z^{- 2}$

Solution walkthrough

The product of $(x^{- 6} y^{3} z^{5})$ and $(x^{4} z^{5} + y^{8} z^{- 7})$ can be represented by the expression $(x^{- 6} y^{3} z^{5}) (x^{4} z^{5}) + (x^{- 6} y^{3} z^{5}) (y^{8} z^{- 7})$ , or $x^{- 6 + 4} y^{3} z^{5 + 5} + x^{- 6} y^{3} y^{8} z^{5 - 7}$ .
This expression is equivalent to $x^{- 2} y^{3} z^{10} + x^{- 6} y^{11} z^{- 2}$ .

Question 2 of 3Hard

For what value of x is the given expression equivalent to (70n)^{30x}, where n > 1?

5 70 n (70 n)^{2}

$\frac{1}{150}$
$\frac{1}{30}$
$\frac{1}{25}$
$\frac{6}{5}$

Show answer and solution

Correct answer $C . \frac{1}{25}$

Solution walkthrough

The correct answer is $\frac{1}{25}$ . An expression of the form $k a$ , where $k$ is an integer greater than 1 and $a \geq 0$ , is equivalent to $a^{\frac{1}{k}}$ .
Therefore, the given expression, where $n > 1$ , is equivalent to $(70 n)^{\frac{1}{5}} ((70 n)^{\frac{1}{2}})^{2}$ . Applying properties of exponents, this expression can be rewritten as $(70 n)^{\frac{1}{5}} (70 n)^{\frac{1}{2} \cdot 2}$ , or $(70 n)^{\frac{1}{5}} (70 n)^{1}$ , which can be rewritten as $(70 n)^{\frac{1}{5} + 1}$ , or $(70 n)^{\frac{6}{5}}$ .
It's given that the expression $5 70 n (70 n)^{2}$ is equivalent to $(70 n)^{30 x}$ , where $n > 1$ . It follows that $(70 n)^{\frac{6}{5}}$ is equivalent to $(70 n)^{30 x}$ .
Therefore, $\frac{6}{5} = 30 x$ . Dividing both sides of this equation by 30 yields $x = \frac{6}{150}$ , which simplifies to $\frac{1}{25}$ .
Thus, the value of $x$ is $\frac{1}{25}$ .

Question 3 of 3Hard

\frac{x ^{5}}{3 x ^{4}} = x^{\frac{a}{b}}

for all positive values of x, what is the value of

\frac{a}{b}

$\frac{7}{6}$
$\frac{1}{3}$
$\frac{4}{3}$
$\frac{3}{2}$

Show answer and solution

Correct answer $A . \frac{7}{6}$

Solution walkthrough

Rewrite the radicals using rational exponents: $x^{5} = x^{5/2}$ and $3 x^{4} = x^{4/3}$ .
Then use the quotient rule for exponents: $x^{5/2} / x^{4/3} = x^{5/2 - 4/3}$ .
Subtracting gives $\frac{15}{6} - \frac{8}{6} = \frac{7}{6}$ , so $x^{5/2 - 4/3} = x^{7/6}$ .
Therefore, $\frac{a}{b} = \frac{7}{6}$ , which is choice A.

Why the other choices miss

Choice B is incorrect because it comes from subtracting the exponents incorrectly. Choice C is incorrect because it comes from adding instead of subtracting. Choice D is incorrect because it does not match the simplified exponent.

Avoid these traps

Common mistakes on this skill

Adding exponents during addition

Exponents add when multiplying powers with the same base, not when adding the powers themselves.

Applying a power to only one factor

A power outside parentheses applies to every factor inside and multiplies existing exponents.

Study plan

How to practice this skill in Dolphin

Identify whether the operation is multiplication, division, or a power of a power.
Apply one exponent rule at a time while keeping the base unchanged where required.
Rewrite negative or fractional exponents and check the result with a simple positive base.

Practice exponent rules in Dolphin SAT

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FAQ

Questions about SAT Exponent Rules Practice

What does a negative exponent mean?

For a nonzero base, a^-n equals 1 divided by a^n.

When do exponents multiply?

They multiply in a power of a power, such as (x^a)^b = x^(ab).