Calculate the Negative Third Power of a Compound Fraction

Q: Why do I need to flip the fraction when the exponent is negative?

A negative exponent means "take the reciprocal and make the exponent positive." So $ \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n $. It's like saying "flip it upside down!"

Q: Should I multiply 9 × 7 = 63 before applying the exponent?

Either way works! You can work with $ \left(\frac{63}{4}\right)^3 $ or keep it as $ \frac{9^3 \times 7^3}{4^3} $. The second form often matches answer choices better.

Q: What's the difference between (a/b)^-3 and a^-3/b^3?

They're completely different! $ \left(\frac{a}{b}\right)^{-3} = \frac{b^3}{a^3} $ but $ \frac{a^{-3}}{b^3} = \frac{1}{a^3 \times b^3} $. The first flips the fraction, the second doesn't.

Q: Can I leave my answer with numbers instead of variables?

Check the answer choices! If they show $ 9^3 \times 7^3 $, keep it that way. If they want $ 63^3 $, then multiply first. Match the format expected.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 According to the laws of exponents, a fraction that is raised to a negative power (-N)

00:09 is equal to the reciprocal raised to the opposite power (N)

00:12 We will apply this formula to our exercise

00:16 We will convert to the reciprocal and raise to the opposite power

00:29 According to the laws of exponents, a fraction raised to the power (N)

00:33 is equal to the fraction where both the numerator and denominator are raised to the power (N)

00:36 We will apply this formula to our exercise

00:40 We will raise both the numerator and denominator to the appropriate power, maintaining the parentheses

00:43 According to the laws of exponents, a product raised to the power (N)

00:47 is equal to the product broken down into factors where each factor is raised to the power (N)

00:50 We will apply this formula to our exercise

00:53 We will break down each product into factors and raise them to the appropriate power

00:57 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Insert the corresponding expression:

$\left(\frac{4}{9\times7}\right)^{-3}=$

2

Step-by-step solution

Let's solve the problem step-by-step:

The given expression is $\left(\frac{4}{9 \times 7}\right)^{-3}$ .

Step 1: Apply the negative exponent rule. A negative exponent $-n$ can be transformed by reciprocal and changing the sign of the exponent. Therefore, $\left(\frac{4}{9 \times 7}\right)^{-3} = \left(\frac{9 \times 7}{4}\right)^3$ .

Step 2: Express the denominator as a product of integers for clarity: $\frac{9 \times 7}{4}$ is clearer than $\frac{63}{4}$ in context for further steps.

Step 3: Apply the power of a fraction rule. Where $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ , therefore, $\left(\frac{9 \times 7}{4}\right)^3 = \frac{(9 \times 7)^3}{4^3}$ .

Step 4: Separate the powers within the fraction: The expression $(9 \times 7)^3$ can be expanded using individual exponents: $(9 \times 7)^3 = 9^3 \times 7^3$ .

Thus, our expression simplifies to: $\frac{9^3 \times 7^3}{4^3}$ .

After analyzing the problem and solving it using the rules of exponents, we conclude that the correct expression is $\frac{9^3 \times 7^3}{4^3}$ .

Therefore, the correct choice from the provided options is choice 2: $\frac{9^3 \times 7^3}{4^3}$ .

3

Final Answer

$\frac{9^3\times7^3}{4^3}$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent means flip the fraction and make exponent positive
Technique: $\left(\frac{4}{63}\right)^{-3} = \left(\frac{63}{4}\right)^3$ by reciprocal rule
Check: Apply power to numerator and denominator separately: $\frac{9^3 \times 7^3}{4^3}$ ✓

Common Mistakes

Avoid these frequent errors

Applying negative exponent only to numerator
Don't just make 4 negative like $\frac{4^{-3}}{9^3 \times 7^3}$ = wrong answer! This ignores the reciprocal rule for fractions. Always flip the entire fraction first, then apply the positive exponent to both parts.

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 flip the fraction when the exponent is negative?

+

A negative exponent means "take the reciprocal and make the exponent positive." So $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ . It's like saying "flip it upside down!"

Should I multiply 9 × 7 = 63 before applying the exponent?

+

Either way works! You can work with $\left(\frac{63}{4}\right)^3$ or keep it as $\frac{9^3 \times 7^3}{4^3}$ . The second form often matches answer choices better.

What's the difference between (a/b)^-3 and a^-3/b^3?

+

They're completely different! $\left(\frac{a}{b}\right)^{-3} = \frac{b^3}{a^3}$ but $\frac{a^{-3}}{b^3} = \frac{1}{a^3 \times b^3}$ . The first flips the fraction, the second doesn't.

How do I know which answer choice is correct?

+

Look for the pattern: original fraction flipped + positive exponent applied to both parts. In our case, $\frac{4}{9 \times 7}$ becomes $\frac{9^3 \times 7^3}{4^3}$ .

Can I leave my answer with numbers instead of variables?

+

Check the answer choices! If they show $9^3 \times 7^3$ , keep it that way. If they want $63^3$ , then multiply first. Match the format expected.

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Calculate the Negative Third Power of a Compound Fraction

Negative Exponents with Compound Fractions

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