Solve the Fraction: Evaluating 1/4^(-3) Step by Step

Negative Exponents with Fraction Reciprocals

143=? \frac{1}{4^{-3}}=?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Solve the following problem
00:03 According to the exponent laws, a number(A) when raised to the power of(-N)
00:06 equals 1 divided by the number(A) raised to the power of(N)
00:09 Let's apply this to the question, the formula works from the number to fraction and vice versa
00:12 We obtain the number(4) raised to the power of-(-3)
00:15 A negative multiplied by a negative always equals a positive
00:19 Let's calculate 4 raised to the power of 3 according to the exponent laws
00:23 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

143=? \frac{1}{4^{-3}}=?

2

Step-by-step solution

First let's recall the negative exponent rule:

an=1an a^{-n}=\frac{1}{a^n} We'll apply it to the expression we received:

143=4(3)=43=64 \frac{1}{4^{-3}}=4^{-(-3)}=4^3=64 In the first stage, we carefully applied the above exponent rule, and since the term in the denominator is already a negative exponent, when using the mentioned rule we put the exponent of the term that was in the denominator in parentheses (this is to apply the minus sign associated with the exponent rule later), then we simplified the exponent expression that was obtained.

In the final stage, we calculated the actual numerical result of the expression we received.

Therefore, the correct answer is answer B.

3

Final Answer

64 64

Key Points to Remember

Essential concepts to master this topic
  • Rule: Negative exponent means take the reciprocal: an=1an a^{-n} = \frac{1}{a^n}
  • Technique: 143=4(3)=43=64 \frac{1}{4^{-3}} = 4^{-(-3)} = 4^3 = 64
  • Check: Verify that 43=164 4^{-3} = \frac{1}{64} , so 1164=64 \frac{1}{\frac{1}{64}} = 64

Common Mistakes

Avoid these frequent errors
  • Applying negative exponent rule incorrectly to the fraction
    Don't calculate 143=1164 \frac{1}{4^{-3}} = \frac{1}{\frac{1}{64}} as 164 \frac{1}{64} ! This ignores that dividing by a fraction means multiplying by its reciprocal. Always remember: when you have 143 \frac{1}{4^{-3}} , flip the negative exponent to get 43 4^3 .

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why does a negative exponent in the denominator become positive?

+

When you have 1an \frac{1}{a^{-n}} , you're dividing by a fraction! Since an=1an a^{-n} = \frac{1}{a^n} , dividing by 1an \frac{1}{a^n} means multiplying by its reciprocal an a^n .

How do I remember which way the exponent changes?

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Think of it as "flipping twice": First, the negative exponent flips the base to the denominator. Then, moving from denominator to numerator flips it back, making the exponent positive!

What if I calculated 43 4^{-3} first?

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That works too! 43=164 4^{-3} = \frac{1}{64} , so 143=1164=64 \frac{1}{4^{-3}} = \frac{1}{\frac{1}{64}} = 64 . Just remember that dividing by a fraction means multiplying by its reciprocal.

Why isn't the answer negative since there's a negative exponent?

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The negative sign is in the exponent, not in front of the whole expression! 43 4^{-3} means "take the reciprocal", not "make it negative". The result is always positive when the base is positive.

Can I use this method for any negative exponent in a denominator?

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Yes! The rule 1an=an \frac{1}{a^{-n}} = a^n works for any positive base and any integer exponent. Just flip the negative exponent to positive when moving to the numerator.

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