Calculate (5×7/71)⁴: Solving a Complex Fraction Power

Q: Why do both the numerator and denominator get the exponent?

Because of the power of a fraction rule! When you raise a fraction to a power, it's like multiplying the fraction by itself that many times. So $ \left(\frac{a}{b}\right)^4 = \frac{a}{b} \times \frac{a}{b} \times \frac{a}{b} \times \frac{a}{b} = \frac{a^4}{b^4} $.

Q: Do I need to calculate 5×7 first before applying the exponent?

No! You can leave it as $ (5\times7)^4 $ in the numerator. This keeps the expression in its exact form, which is often what math problems ask for.

Q: What's the difference between (5×7)⁴ and 5×7⁴?

Huge difference! $ (5\times7)^4 $ means multiply 5 and 7 first, then raise to the 4th power. But $ 5\times7^4 $ means raise only 7 to the 4th power, then multiply by 5. Order of operations matters!

Q: Why can't I distribute the exponent like 5⁴×7⁴/71⁴?

You can do that! $ (5\times7)^4 = 5^4\times7^4 $ because of the power of a product rule. But the question asks for the form that matches the original structure, which keeps $ (5\times7)^4 $ together.

Q: How do I remember the power of a fraction rule?

Think of it as "exponent goes everywhere" - when you see $ \left(\frac{\text{top}}{\text{bottom}}\right)^n $, the exponent affects both the top AND bottom parts!

Fraction Exponents with Product Numerators

Insert the corresponding expression:

$\left(\frac{5\times7}{71}\right)^4=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's simplify this problem together.

00:12 When a fraction is raised to a power, like N, it means both the top, and bottom, go to that power.

00:18 So, each part of the fraction gets that exponent, N.

00:23 Now, we'll use this idea to solve our problem.

00:27 Notice, there's a product on top. Be sure to use parentheses correctly.

00:32 Great job! And that's how you find the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{5\times7}{71}\right)^4=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the expression inside the parentheses that needs to be exponentiated.
Step 2: Apply the power of a fraction rule, $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .
Step 3: Compute the expression based on this rule.

Now, let's work through each step:

Step 1: The initial expression provided is $\left(\frac{5 \times 7}{71}\right)$ . The numerator is $5 \times 7$ , and the denominator is $71$ .

Step 2: According to the power of a fraction rule, we apply the exponent $4$ to both the numerator and the denominator:

$\left(\frac{5 \times 7}{71}\right)^4 = \frac{(5 \times 7)^4}{71^4}$ .

Step 3: Simply ensure the expression aligns with the given multiple-choice options.

The expression is, therefore, $\frac{(5 \times 7)^4}{71^4}$ , which matches the choice:

$\frac{\left(5\times7\right)^4}{71^4}$ (Choice 3).

Thus, the solution to the problem is correctly matched as $\frac{\left(5\times7\right)^4}{71^4}$ .

Final Answer

$\frac{\left(5\times7\right)^4}{71^4}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Apply exponent to both numerator and denominator separately
Technique: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ means both parts get the exponent
Check: Ensure parentheses group numerator correctly: (5×7)⁴ not 5×7⁴ ✓

Common Mistakes

Avoid these frequent errors

Applying exponent only to denominator
Don't write $\frac{5\times7}{71^4}$ = wrong distribution! This ignores the power rule for fractions and gives an incorrect expression. Always apply the exponent to both the entire numerator and denominator: $\left(\frac{5\times7}{71}\right)^4 = \frac{(5\times7)^4}{71^4}$ .

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why do both the numerator and denominator get the exponent?

Because of the power of a fraction rule! When you raise a fraction to a power, it's like multiplying the fraction by itself that many times. So $\left(\frac{a}{b}\right)^4 = \frac{a}{b} \times \frac{a}{b} \times \frac{a}{b} \times \frac{a}{b} = \frac{a^4}{b^4}$ .

Do I need to calculate 5×7 first before applying the exponent?

No! You can leave it as $(5\times7)^4$ in the numerator. This keeps the expression in its exact form, which is often what math problems ask for.

What's the difference between (5×7)⁴ and 5×7⁴?

Huge difference! $(5\times7)^4$ means multiply 5 and 7 first, then raise to the 4th power. But $5\times7^4$ means raise only 7 to the 4th power, then multiply by 5. Order of operations matters!

Why can't I distribute the exponent like 5⁴×7⁴/71⁴?

You can do that! $(5\times7)^4 = 5^4\times7^4$ because of the power of a product rule. But the question asks for the form that matches the original structure, which keeps $(5\times7)^4$ together.

How do I remember the power of a fraction rule?

Think of it as "exponent goes everywhere" - when you see $\left(\frac{\text{top}}{\text{bottom}}\right)^n$ , the exponent affects both the top AND bottom parts!

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