Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\left(\frac{1}{4}\right)^{-2} = 16
\]
Solution
To evaluate the expression \( \left(\frac{1}{4}\right)^{-2} \), let's break it down step by step using the rules of exponents.
### Step 1: Understand Negative Exponents
A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. In general:
\[
a^{-n} = \frac{1}{a^n}
\]
So,
\[
\left(\frac{1}{4}\right)^{-2} = \frac{1}{\left(\frac{1}{4}\right)^2}
\]
### Step 2: Calculate the Positive Exponent
First, compute \( \left(\frac{1}{4}\right)^2 \):
\[
\left(\frac{1}{4}\right)^2 = \frac{1^2}{4^2} = \frac{1}{16}
\]
### Step 3: Take the Reciprocal
Now, take the reciprocal of \( \frac{1}{16} \):
\[
\frac{1}{\left(\frac{1}{4}\right)^2} = \frac{1}{\frac{1}{16}} = 16
\]
### Final Answer
\[
\left(\frac{1}{4}\right)^{-2} = 16
\]
Reviewed and approved by the UpStudy tutoring team
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Explain 
Simplify this solution Mind Expander
When you encounter an expression like \( \left(\frac{1}{4}\right)^{-2} \), you can simplify it using the rule of negative exponents. A negative exponent indicates that you should take the reciprocal of the base, so \( \left(\frac{1}{4}\right)^{-2} \) becomes \( \left(4\right)^{2} \). Thus, calculating \( 4^2 \) gives you an answer of 16! A common mistake people make is forgetting that a negative exponent means flipping the fraction. Instead of thinking \( \left(\frac{1}{4}\right)^{-2} \) equals \( \frac{1}{4^2} \), remember to take the reciprocal first! It's crucial to keep track of those negative signs to avoid any mix-ups during calculations.
