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Apply All Exponent Rules Practice

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Question
Apply the Properties of Integer Exponents to generate equivalent expressions to 4^0 ⋅ 5^−2/(−2)^−3
with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
4^0 ⋅ 5^−2/(−2)^−3=

1 answer

To simplify the given expression \( \frac{4^0 \cdot 5^{-2}}{(-2)^{-3}} \) using the properties of integer exponents, follow these steps:

  1. Simplify \(4^0\): Any non-zero number raised to the power of 0 is 1. Thus, \(4^0 = 1\).

    \[ \frac{1 \cdot 5^{-2}}{(-2)^{-3}} \]

  2. Rewrite \(5^{-2}\): According to the negative exponent rule, \(a^{-n} = \frac{1}{a^n}\). So,

    \[ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \]

    Now we can update our expression:

    \[ \frac{1/25}{(-2)^{-3}} \]

  3. Rewrite \((-2)^{-3}\): Similarly,

    \[ (-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8} \]

    Now we can rewrite the expression:

    \[ \frac{1/25}{1/(-8)} = \frac{1}{25} \cdot \frac{-8}{1} = \frac{-8}{25} \]

  4. Final simplified result:

    \[ \frac{-8}{25} \]

Thus, the final answer is:

\[ \frac{-8}{25} \]