Question
Simplify the expression. \( 7^{-2} \) -49 \( (-7)^{2} \) \( -7^{2} \) \( \frac{1}{49} \)
Ask by Salinas West. in the United States
Jan 08,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( 7^{-2} = \frac{1}{49} \)
Solution
To simplify the expression \( 7^{-2} \), let's break it down step by step.
### Step 1: Understanding Negative Exponents
A negative exponent indicates the reciprocal of the base raised to the positive exponent. In general:
\[
a^{-n} = \frac{1}{a^n}
\]
### Step 2: Apply to \( 7^{-2} \)
Using the rule above:
\[
7^{-2} = \frac{1}{7^2}
\]
### Step 3: Compute \( 7^2 \)
\[
7^2 = 7 \times 7 = 49
\]
### Step 4: Final Simplification
\[
7^{-2} = \frac{1}{49}
\]
### Summary of Related Expressions
1. **\( 7^{-2} \)**
\[
7^{-2} = \frac{1}{49}
\]
2. **\(-49\)**
This is simply \(-49\).
3. **\( (-7)^2 \)**
\[
(-7)^2 = (-7) \times (-7) = 49
\]
4. **\( -7^{2} \)**
Be cautious with the placement of the negative sign. This expression is interpreted as:
\[
-7^{2} = - (7 \times 7) = -49
\]
5. **\( \frac{1}{49} \)**
This is already in its simplest form and equals \( 7^{-2} \).
### Conclusion
The simplified form of \( 7^{-2} \) is:
\[
\boxed{\dfrac{1}{49}}
\]
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Simplify this solution Bonus Knowledge
To simplify \( 7^{-2} \), we can recall that a negative exponent means we take the reciprocal of the base raised to the positive exponent. Thus, \( 7^{-2} = \frac{1}{7^{2}} \). Since \( 7^{2} = 49 \), we conclude that \( 7^{-2} = \frac{1}{49} \). It’s crucial to remember that negative exponents lead to reciprocation. A common mistake here would be misinterpreting it as simply squaring the base and applying the negative sign, resulting in confusion. Always flip that base for proper simplification!
