Solve (4/35)^4: Evaluating a Compound Fraction to the Fourth Power

Q: Why do I need parentheses around the whole denominator?

The parentheses show that both 5 and 7 get raised to the 4th power. Without parentheses, only the number closest to the exponent gets the power, giving you a wrong answer!

Q: Can I simplify 5×7 to 35 first?

Absolutely! You can write it as $ \left(\frac{4}{35}\right)^4 = \frac{4^4}{35^4} $. Both approaches are correct - use whichever feels easier to you.

Q: Do I have to calculate 4⁴ and (5×7)⁴?

Not necessarily! The question asks for the expression, not the final numerical answer. $ \frac{4^4}{(5\times7)^4} $ is the complete correct form.

Q: What if the fraction had addition or subtraction in the denominator?

Same rule applies! For $ \left(\frac{3}{5+2}\right)^2 $, you'd get $ \frac{3^2}{(5+2)^2} $. The entire denominator always gets the exponent.

Q: How do I remember this exponent rule?

Think of it as: "What goes up, must come up!" If the whole fraction is raised to a power, then The top (numerator) gets that powerThe bottom (denominator) gets that same power

Fraction Exponents with Product Denominators

Insert the corresponding expression:

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

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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 laws of exponents, a fraction raised to the power (N)

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

00:11 We will apply this formula to our exercise

00:20 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:

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

Step-by-step solution

To simplify the expression $\left(\frac{4}{5 \times 7}\right)^4$ , we will apply the rule of exponents for fractions, which states that $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

Step 1: Identify the components of the fraction.
The fraction is $\frac{4}{5 \times 7}$ .

Step 2: Apply the exponent to both the numerator and the denominator separately.
The numerator is 4, and the calculation is $4^4$ .
The denominator is $5 \times 7$ , and the calculation is $(5 \times 7)^4$ .

Step 3: Write the expression with the exponents.
The simplified expression becomes $\frac{4^4}{(5 \times 7)^4}$ .

Therefore, the correct expression is $\frac{4^4}{(5 \times 7)^4}$ , which corresponds to choice 1.

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Rule: Apply exponents to both numerator and denominator separately
Technique: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ so $\left(\frac{4}{5\times7}\right)^4 = \frac{4^4}{(5\times7)^4}$
Check: Verify both parts have exponent 4: numerator becomes $4^4$ , denominator becomes $(5\times7)^4$ ✓

Common Mistakes

Avoid these frequent errors

Applying exponent only to part of the denominator
Don't write $\frac{4^4}{5\times7^4}$ or $\frac{4^4}{5\times7}$ = wrong result! The exponent must apply to the entire denominator as one unit. Always use parentheses: $(5\times7)^4$ to show the whole denominator gets the exponent.

Practice Quiz

Test your knowledge with interactive questions

$5^4\times25=$

FAQ

Everything you need to know about this question

Why do I need parentheses around the whole denominator?

The parentheses show that both 5 and 7 get raised to the 4th power. Without parentheses, only the number closest to the exponent gets the power, giving you a wrong answer!

Can I simplify 5×7 to 35 first?

Absolutely! You can write it as $\left(\frac{4}{35}\right)^4 = \frac{4^4}{35^4}$ . Both approaches are correct - use whichever feels easier to you.

Do I have to calculate 4⁴ and (5×7)⁴?

Not necessarily! The question asks for the expression, not the final numerical answer. $\frac{4^4}{(5\times7)^4}$ is the complete correct form.

What if the fraction had addition or subtraction in the denominator?

Same rule applies! For $\left(\frac{3}{5+2}\right)^2$ , you'd get $\frac{3^2}{(5+2)^2}$ . The entire denominator always gets the exponent.

How do I remember this exponent rule?

Think of it as: "What goes up, must come up!" If the whole fraction is raised to a power, then

The top (numerator) gets that power
The bottom (denominator) gets that same power

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