Evaluate (6/x)³: Complete the Cube of a Fraction Expression

Q: Why do I need to apply the exponent to both the top and bottom?

The power of a fraction rule says that when you raise a fraction to an exponent, you must apply that exponent to both the numerator and denominator. Think of it as: $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $

Q: What if I forget to put the exponent on the denominator?

You'll get the wrong answer! For example, $ \frac{6^3}{x} = \frac{216}{x} $ is completely different from $ \frac{6^3}{x^3} = \frac{216}{x^3} $. Always check that both parts have the exponent.

Q: Do I need to calculate 6³ or can I leave it as 6³?

Both $ \frac{6^3}{x^3} $ and $ \frac{216}{x^3} $ are correct! The question asks for the expression form, so 6³ in the numerator is perfectly acceptable.

Q: Does this rule work for any exponent?

Yes! Whether it's $ \left(\frac{a}{b}\right)^2 $, $ \left(\frac{a}{b}\right)^4 $, or any other exponent, you always apply it to both the numerator and denominator separately.

Q: What about negative exponents like (6/x)⁻³?

The same rule applies! $ \left(\frac{6}{x}\right)^{-3} = \frac{6^{-3}}{x^{-3}} $, which can be simplified further using negative exponent rules.

Exponent Rules with Fractional Bases

Insert the corresponding expression:

$\left(\frac{6}{x}\right)^3=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the exponent laws, a fraction raised to the power (N)

00:07 equals the numerator and denominator raised to the same power (N)

00:10 We will apply this formula to our exercise

00:14 We will raise both numerator and denominator to the power (N)

00:17 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{6}{x}\right)^3=$

Step-by-step solution

To solve this problem, we will rewrite the expression $\left(\frac{6}{x}\right)^3$ using the power of a fraction rule. The steps are as follows:

Identify the fraction's numerator $6$ and denominator $x$ .
According to the power of a fraction rule, apply the power 3 to both the numerator and the denominator:
$\left(\frac{6}{x}\right)^3 = \frac{6^3}{x^3}$ .

Therefore, the expression is correctly written as $\frac{6^3}{x^3}$ .

Comparing with the provided answer choices, the correct choice is choice $2$ :

$\frac{6^3}{x^3}$

Final Answer

$\frac{6^3}{x^3}$

Key Points to Remember

Essential concepts to master this topic

Power of Fraction Rule: Apply exponent to both numerator and denominator
Technique: $\left(\frac{6}{x}\right)^3 = \frac{6^3}{x^3}$ distributes power to both parts
Check: Verify both numerator and denominator have the exponent applied ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only one part of the fraction
Don't apply the exponent to just the numerator: $\left(\frac{6}{x}\right)^3 \neq \frac{6^3}{x}$ ! This ignores the power rule for fractions and gives an incomplete result. Always apply the exponent to both the numerator AND denominator separately.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I need to apply the exponent to both the top and bottom?

The power of a fraction rule says that when you raise a fraction to an exponent, you must apply that exponent to both the numerator and denominator. Think of it as: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

What if I forget to put the exponent on the denominator?

You'll get the wrong answer! For example, $\frac{6^3}{x} = \frac{216}{x}$ is completely different from $\frac{6^3}{x^3} = \frac{216}{x^3}$ . Always check that both parts have the exponent.

Do I need to calculate 6³ or can I leave it as 6³?

Both $\frac{6^3}{x^3}$ and $\frac{216}{x^3}$ are correct! The question asks for the expression form, so 6³ in the numerator is perfectly acceptable.

Does this rule work for any exponent?

Yes! Whether it's $\left(\frac{a}{b}\right)^2$ , $\left(\frac{a}{b}\right)^4$ , or any other exponent, you always apply it to both the numerator and denominator separately.

What about negative exponents like (6/x)⁻³?

The same rule applies! $\left(\frac{6}{x}\right)^{-3} = \frac{6^{-3}}{x^{-3}}$ , which can be simplified further using negative exponent rules.

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