Evaluate 6 to the power of -2
The Correct Answer and Explanation is:
To evaluate ( 6^{-2} ), we need to understand the rule for negative exponents. The general rule for exponents states that:
[
a^{-n} = \frac{1}{a^n}
]
Where ( a ) is the base and ( n ) is the exponent. So, for ( 6^{-2} ), we apply this rule:
[
6^{-2} = \frac{1}{6^2}
]
Now, we evaluate ( 6^2 ), which means multiplying 6 by itself:
[
6^2 = 6 \times 6 = 36
]
Therefore:
[
6^{-2} = \frac{1}{36}
]
So, the value of ( 6^{-2} ) is ( \frac{1}{36} ).
Detailed Explanation:
Exponents represent repeated multiplication. A positive exponent means multiplying the base by itself that many times, while a negative exponent indicates the reciprocal (or inverse) of that number raised to the positive exponent. In this case, the negative exponent of -2 tells us to take the reciprocal of ( 6^2 ), which results in ( \frac{1}{36} ).
To clarify further:
- ( 6^2 ) is simply 6 multiplied by itself, which gives 36.
- ( 6^{-2} ) reverses this, giving ( \frac{1}{36} ). This is an important concept because it shows how negative exponents are used to express values smaller than 1.
In practical terms, when you encounter a negative exponent in mathematics, it’s useful to remember that it means you’re dealing with the reciprocal of the base raised to the positive exponent. Negative exponents are very common in algebra, calculus, and other advanced mathematical topics, and they help express very small numbers in a compact form.
Thus, the final result of ( 6^{-2} ) is ( \frac{1}{36} ).