Evaluate (3/7)^(-4): Negative Exponent Expression

Negative Exponents with Fractional Bases

Insert the corresponding expression:

(37)−4= \left(\frac{3}{7}\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
00:06 According to the power laws, a fraction raised to a power (N)
00:09 equals the numerator and denominator, raised to the same power(N)
00:14 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
1

Understand the problem

Insert the corresponding expression:

(37)−4= \left(\frac{3}{7}\right)^{-4}=

3

Final Answer

3−47−4 \frac{3^{-4}}{7^{-4}}

Key Points to Remember

Essential concepts to master this topic
  • Rule: Apply negative exponent to both numerator and denominator
  • Technique: (37)−4=3−47−4 \left(\frac{3}{7}\right)^{-4} = \frac{3^{-4}}{7^{-4}} keeps the fraction structure
  • Check: Verify that 3−47−4=(73)4 \frac{3^{-4}}{7^{-4}} = \left(\frac{7}{3}\right)^4 by flipping ✓

Common Mistakes

Avoid these frequent errors
  • Only applying negative exponent to numerator or denominator
    Don't write 3−47 \frac{3^{-4}}{7} or 37−4 \frac{3}{7^{-4}} = partial application gives wrong answer! The negative exponent affects the entire fraction base. Always apply the exponent to both parts: (37)−4=3−47−4 \left(\frac{3}{7}\right)^{-4} = \frac{3^{-4}}{7^{-4}} .

Practice Quiz

Test your knowledge with interactive questions

\( 112^0=\text{?} \)

FAQ

Everything you need to know about this question

Why does the negative exponent apply to both 3 and 7?

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The negative exponent applies to the entire base 37 \frac{3}{7} . When you have (ab)n \left(\frac{a}{b}\right)^n , the exponent affects both parts: anbn \frac{a^n}{b^n} .

Can I write this as a positive exponent instead?

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Yes! (37)−4 \left(\frac{3}{7}\right)^{-4} equals (73)4 \left(\frac{7}{3}\right)^4 . The negative exponent flips the fraction and makes the exponent positive.

Why isn't the answer negative since the exponent is negative?

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A negative exponent doesn't make the result negative! It means reciprocal (flip the fraction). The negative sign affects the exponent rule, not the final answer's sign.

How do I simplify 3−47−4 \frac{3^{-4}}{7^{-4}} further?

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Use the rule a−nb−n=bnan \frac{a^{-n}}{b^{-n}} = \frac{b^n}{a^n} . So 3−47−4=7434=(73)4 \frac{3^{-4}}{7^{-4}} = \frac{7^4}{3^4} = \left(\frac{7}{3}\right)^4 .

What if I see a negative sign outside the parentheses?

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Be careful! −(73)4 -\left(\frac{7}{3}\right)^4 has a negative sign in front, making the final answer negative. This is different from our expression which has no negative sign outside.

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