To simplify the expression \( \frac{4^0 \times 5^{-2}}{-2^{-3}} \), we will apply the properties of exponents:
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Any number to the power of 0 is 1: \[ 4^0 = 1 \]
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The property of negative exponents states that \( a^{-n} = \frac{1}{a^n} \): \[ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \]
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The negative exponent in the denominator can be adjusted: \[ -2^{-3} = -\frac{1}{2^3} = -\frac{1}{8} \]
Now substitute these results back into the expression: \[ \frac{4^0 \times 5^{-2}}{-2^{-3}} = \frac{1 \times \frac{1}{25}}{-\frac{1}{8}} = \frac{\frac{1}{25}}{-\frac{1}{8}} \]
When dividing fractions, we can multiply by the reciprocal of the denominator: \[ = \frac{1}{25} \times -8 = -\frac{8}{25} \]
Thus, the simplified expression for \( \frac{4^0 \times 5^{-2}}{-2^{-3}} \) is: \[ -\frac{8}{25} \]