Evaluate (1/4)^(-1): Converting Negative Exponents to Reciprocals

Negative Exponents with Reciprocal Fractions

(14)−1 (\frac{1}{4})^{-1}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:04 Let's simplify this together.
00:07 We're using a formula for negative exponents.
00:11 If a fraction, A over B, has a negative exponent, negative M,
00:16 we take the reciprocal, B over A, and use a positive exponent, M.
00:22 Let's try this technique in our exercise.
00:25 We'll switch to the reciprocal and change the exponent to positive.
00:30 Remember, any number to the power of one is just that number.
00:34 And that's how we solve the problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

(14)−1 (\frac{1}{4})^{-1}

2

Step-by-step solution

We use the power property for a negative exponent:

a−n=1an a^{-n}=\frac{1}{a^n} We will write the fraction in parentheses as a negative power with the help of the previously mentioned power:

14=141=4−1 \frac{1}{4}=\frac{1}{4^1}=4^{-1} We return to the problem, where we obtained:

(14)−1=(4−1)−1 \big(\frac{1}{4}\big)^{-1}=(4^{-1})^{-1} We continue and use the power property of an exponent raised to another exponent:

(am)n=am⋅n (a^m)^n=a^{m\cdot n} And we apply it in the problem:

(4−1)−1=4−1⋅−1=41=4 (4^{-1})^{-1}=4^{-1\cdot-1}=4^1=4 Therefore, the correct answer is option d.

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Final Answer

4 4

Key Points to Remember

Essential concepts to master this topic
  • Rule: Negative exponents create reciprocals: a−n=1an a^{-n} = \frac{1}{a^n}
  • Technique: Rewrite 14=4−1 \frac{1}{4} = 4^{-1} then apply (4−1)−1=41=4 (4^{-1})^{-1} = 4^1 = 4
  • Check: Verify that (14)−1×14=1 (\frac{1}{4})^{-1} \times \frac{1}{4} = 1 which gives 4×14=1 4 \times \frac{1}{4} = 1 ✓

Common Mistakes

Avoid these frequent errors
  • Thinking negative exponents make numbers negative
    Don't assume (14)−1=−14 (\frac{1}{4})^{-1} = -\frac{1}{4} = wrong sign! Negative exponents only flip the base to its reciprocal, they don't change the sign of the result. Always remember: negative exponent means reciprocal, not negative number.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why doesn't the negative exponent make the answer negative?

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Great question! The negative exponent only tells you to flip the base (take its reciprocal). It doesn't affect the sign of your final answer. Think of it as an instruction: 'flip this fraction' not 'make this negative'.

How do I remember what a−n a^{-n} means?

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Remember: negative exponent = reciprocal! The negative sign is like saying 'flip it upside down'. So (14)−1 (\frac{1}{4})^{-1} becomes 41=4 \frac{4}{1} = 4 .

What's the difference between (14)−1 (\frac{1}{4})^{-1} and −(14)1 -(\frac{1}{4})^{1} ?

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Big difference! (14)−1=4 (\frac{1}{4})^{-1} = 4 (positive), but −(14)1=−14 -(\frac{1}{4})^{1} = -\frac{1}{4} (negative). The negative exponent creates a reciprocal, while the negative sign in front makes the result negative.

Can I use this rule with any fraction?

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Yes! For any fraction ab \frac{a}{b} , we have (ab)−1=ba (\frac{a}{b})^{-1} = \frac{b}{a} . Just flip the numerator and denominator. So (37)−1=73 (\frac{3}{7})^{-1} = \frac{7}{3} !

How do I check my answer is correct?

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Multiply your answer by the original base. If you get 1, you're right! For this problem: 4×14=1 4 \times \frac{1}{4} = 1 ✓. This works because any number times its reciprocal equals 1.

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