Evaluate (1/4)^(-2): Negative Exponent Expression

Negative Exponents with Fraction Bases

Insert the corresponding expression:

(14)−2= \left(\frac{1}{4}\right)^{-2}=

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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)
00:10 We will apply this formula to our exercise
00:13 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:

(14)−2= \left(\frac{1}{4}\right)^{-2}=

3

Final Answer

1−24−2 \frac{1^{-2}}{4^{-2}}

Key Points to Remember

Essential concepts to master this topic
  • Rule: Negative exponent means reciprocal: a−n=1an a^{-n} = \frac{1}{a^n}
  • Technique: (14)−2=1(14)2=1116=16 \left(\frac{1}{4}\right)^{-2} = \frac{1}{\left(\frac{1}{4}\right)^2} = \frac{1}{\frac{1}{16}} = 16
  • Check: Verify by rewriting: (14)−2=42=16 \left(\frac{1}{4}\right)^{-2} = 4^2 = 16 ✓

Common Mistakes

Avoid these frequent errors
  • Applying negative exponent to numerator only
    Don't apply the negative exponent to just the numerator: (14)−2≠1−24 \left(\frac{1}{4}\right)^{-2} \neq \frac{1^{-2}}{4} ! This ignores that the entire fraction is the base. Always apply the negative exponent to the whole fraction by taking its reciprocal first.

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

What does a negative exponent actually mean?

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A negative exponent means "take the reciprocal and make the exponent positive." So a−n=1an a^{-n} = \frac{1}{a^n} . It's like flipping the fraction upside down!

Why is the correct answer choice 1−24−2 \frac{1^{-2}}{4^{-2}} ?

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This represents the original expression correctly. When we have (14)−2 \left(\frac{1}{4}\right)^{-2} , we can write it as 1−24−2 \frac{1^{-2}}{4^{-2}} because the exponent applies to both numerator and denominator separately.

How do I simplify 1−24−2 \frac{1^{-2}}{4^{-2}} ?

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Remember that 1−2=112=1 1^{-2} = \frac{1}{1^2} = 1 and 4−2=142=116 4^{-2} = \frac{1}{4^2} = \frac{1}{16} . So 1−24−2=1116=16 \frac{1^{-2}}{4^{-2}} = \frac{1}{\frac{1}{16}} = 16 .

Is there a shortcut for negative exponents with fractions?

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Yes! When you have (ab)−n \left(\frac{a}{b}\right)^{-n} , it equals (ba)n \left(\frac{b}{a}\right)^n . So (14)−2=(41)2=42=16 \left(\frac{1}{4}\right)^{-2} = \left(\frac{4}{1}\right)^2 = 4^2 = 16 !

Why isn't the answer −(14)2 -\left(\frac{1}{4}\right)^2 ?

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The negative sign in the exponent doesn't make the answer negative! It only means "take the reciprocal." A negative exponent is completely different from a negative number.

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