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Apply the Property of Negative Integer Exponents to find the numerical equivalent of 8^-3. The answer should not contain any exponent expression or decimals. Show your work.

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To find the numerical equivalent of \( 8^{-3} \) using the property of negative integer exponents, we can use the following property:

\[ a^{-n} = \frac{1}{a^n} \]

Applying this property to \( 8^{-3} \):

\[ 8^{-3} = \frac{1}{8^3} \]

Now, we need to calculate \( 8^3 \), which means \( 8 \times 8 \times 8 \).

Calculating step by step:

  1. First, calculate \( 8 \times 8 \): \[ 8 \times 8 = 64 \]

  2. Next, multiply the result by \( 8 \): \[ 64 \times 8 = 512 \]

Thus, \( 8^3 = 512 \).

Now substituting this back into our expression:

\[ 8^{-3} = \frac{1}{8^3} = \frac{1}{512} \]

Therefore, the numerical equivalent of \( 8^{-3} \) is

\[ \boxed{\frac{1}{512}} \]

Complete the table. For each expression, choose which exponent rule has been used to simplify. Choose from product rule, quotient rule, power rule, negative integer exponent rule, zero power rule.
11^80/11^22=11^58
6^-12/9^-15=9^15/6^12
(16^3)^9=16^27

Below is the completed table with the exponents and the corresponding exponent rule used for each expression.

| Expression | Exponent Rule | |-------------------------------------|--------------------------------| | \( \frac{11^{80}}{11^{22}} = 11^{58} \) | Quotient Rule | | \( \frac{6^{-12}}{9^{-15}} = \frac{9^{15}}{6^{12}} \) | Negative Integer Exponent Rule | | \( (16^3)^9 = 16^{27} \) | Power Rule |

Explanation of the rules:

  1. Quotient Rule: When you divide two powers with the same base, you subtract the exponents: \( a^m / a^n = a^{m-n} \).

  2. Negative Integer Exponent Rule: When you have negative exponents, it can be expressed as the reciprocal. Specifically, \( a^{-n} = \frac{1}{a^n} \). Hence, \( \frac{6^{-12}}{9^{-15}} = \frac{1}{6^{12}} \cdot \frac{1}{9^{-15}} = \frac{9^{15}}{6^{12}} \).

  3. Power Rule: When raising a power to another power, you multiply the exponents: \( (a^m)^n = a^{m \cdot n} \). Hence, \( (16^3)^9 = 16^{3 \cdot 9} = 16^{27} \).

If you need further clarification on any of the points, feel free to ask!