Write the expression 12^-2 in simplest form.
The Correct Answer and Explanation is:
Correct Answer: 12−2=1122=114412^{-2} = \frac{1}{12^2} = \frac{1}{144}
Explanation
The expression 12−212^{-2} involves a negative exponent, which can seem confusing at first. To simplify it properly, we need to understand what a negative exponent means in mathematics.
What Does a Negative Exponent Mean?
A negative exponent indicates reciprocal. Specifically, for any non-zero number aa and positive integer nn, a−n=1ana^{-n} = \frac{1}{a^n}
This rule tells us that instead of multiplying the base repeatedly (as we do with positive exponents), we divide 1 by the base raised to the positive version of the exponent.
Applying the Rule to 12⁻²:
We use the rule: 12−2=112212^{-2} = \frac{1}{12^2}
Now, calculate 12212^2 (which means 12 × 12): 122=14412^2 = 144
So, 12−2=114412^{-2} = \frac{1}{144}
This is the simplest form of the expression.
Why Does This Rule Make Sense?
The rules for exponents are based on patterns that remain consistent across positive, zero, and negative exponents. For example: 122=144121=12120=112−1=11212−2=114412^2 = 144 \\ 12^1 = 12 \\ 12^0 = 1 \\ 12^{-1} = \frac{1}{12} \\ 12^{-2} = \frac{1}{144}
Each time the exponent decreases by 1, we divide the previous result by 12. So moving from 120=112^0 = 1 to 12−112^{-1}, we divide 1 by 12, getting 112\frac{1}{12}. Then divide again by 12 to get 1144\frac{1}{144}, which is 12−212^{-2}.
Understanding these patterns helps us simplify expressions with negative exponents easily and accurately.
Final Answer: 1144\boxed{\frac{1}{144}}
