Evaluate the Power Fraction: (2^4)/(7^4) Expression

Q: How do I remember the quotient rule for exponents?

Think of it as "same exponent, combine the bases"! When you see $ \frac{a^n}{b^n} $, the exponent n is the same on top and bottom, so you can write it as $ \left(\frac{a}{b}\right)^n $.

Q: What if the exponents were different numbers?

If the exponents are different, you cannot use this rule! The quotient rule $ \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n $ only works when both exponents are exactly the same.

Q: Should I calculate the numerical answer?

For this type of question, you usually want the simplified form $ \left(\frac{2}{7}\right)^4 $ rather than calculating $ \frac{16}{2401} $. The simplified form shows the mathematical relationship more clearly.

Q: Can this rule work backwards too?

Yes! You can go both ways: $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $ and $ \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n $. This flexibility helps you choose the most convenient form for your problem.

Exponent Rules with Fraction Bases

Insert the corresponding expression:

$\frac{2^4}{7^4}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's simplify this problem step by step.

00:09 According to the law of exponents, when a fraction is raised to the power of N,

00:15 both the numerator and denominator are raised to the power of N, too.

00:19 We'll use this formula in reverse to solve our exercise.

00:29 And that's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{2^4}{7^4}=$

Step-by-step solution

To solve this problem, we will apply the exponent rule for powers of a fraction.

Step 1: Understand the given expression $\frac{2^4}{7^4}$ .
Step 2: Use the formula $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ to rewrite the expression.
Step 3: Apply this rule to rewrite $\frac{2^4}{7^4} = \left(\frac{2}{7}\right)^4$ .

This shows that instead of writing separate powers for the numerator and denominator, we can express it as a single fraction raised to that power.

Thus, the expression $\frac{2^4}{7^4}$ corresponds to $\left(\frac{2}{7}\right)^4$ .

The correct choice from the given options is:

Choice 3: $\left(\frac{2}{7}\right)^4$

Therefore, the solution to the problem is $\left(\frac{2}{7}\right)^4$ .

Final Answer

$\left(\frac{2}{7}\right)^4$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same exponent, combine bases
Technique: Transform $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ using quotient rule
Check: Verify $\left(\frac{2}{7}\right)^4 = \frac{2^4}{7^4} = \frac{16}{2401}$ ✓

Common Mistakes

Avoid these frequent errors

Incorrectly multiplying exponent by base numbers
Don't change $\frac{2^4}{7^4}$ to $\frac{2 \times 4}{7 \times 4}$ = wrong calculation! This confuses exponent notation with multiplication and gives incorrect results. Always keep the exponent separate from the base and use the quotient rule $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ .

Practice Quiz

Test your knowledge with interactive questions

$$ Choose the corresponding expression:

$\left(\frac{1}{2}\right)^2=$

FAQ

Everything you need to know about this question

Why can't I just multiply the base by the exponent?

Exponents mean repeated multiplication, not regular multiplication! $2^4$ means 2 × 2 × 2 × 2, not 2 × 4. The exponent tells you how many times to use the base as a factor.

How do I remember the quotient rule for exponents?

Think of it as "same exponent, combine the bases"! When you see $\frac{a^n}{b^n}$ , the exponent n is the same on top and bottom, so you can write it as $\left(\frac{a}{b}\right)^n$ .

What if the exponents were different numbers?

If the exponents are different, you cannot use this rule! The quotient rule $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ only works when both exponents are exactly the same.

Should I calculate the numerical answer?

For this type of question, you usually want the simplified form $\left(\frac{2}{7}\right)^4$ rather than calculating $\frac{16}{2401}$ . The simplified form shows the mathematical relationship more clearly.

Can this rule work backwards too?

Yes! You can go both ways: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ and $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ . This flexibility helps you choose the most convenient form for your problem.

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