Simplify to eliminate all unnecessary exponents. (8^(2)*5^(-4)*8^(5))/(8^(8))

Simplify Exponents: Simplify the expression using the properties of exponents. We have the expression (8^(2)*5^(-4)*8^(5))/(8^(8)). We can combine the powers of 8 by adding the exponents in the numerator and then subtracting the exponent in the denominator.Add Exponents: Add the exponents of 8 in the numerator. 8^(2) * 8^(5) = 8^(2+5) = 8^(7).Divide Powers: Divide the powers of 8 by subtracting the exponents. (8^(7))/(8^(8)) = 8^(7-8) = 8^(-1).Simplify Power: Simplify the power of 5 in the numerator. Since 5^(-4) is in the numerator, it is equivalent to 1/(5^(4)).Combine Powers: Combine the simplified powers of 8 and 5. We have 8^(-1) * 1/(5^(4)). Since 8^(-1) is the same as 1/8, we can rewrite the expression as (1/8) * (1/(5^(4))).Multiply Fractions: Multiply the fractions. (1/8) * (1/(5^(4))) = 1/(8*5^(4)).Calculate Denominator: Calculate the denominator. 8 * 5^(4) = 8 * (5*5*5*5) = 8 * 625 = 5000.Final Expression: Write the final simplified expression. The expression simplifies to 1/5000.

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Precalculus

Operations with rational exponents

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Q. Simplify to eliminate all unnecessary exponents.

\frac{8^{2} \cdot 5^{-4} \cdot 8^{5}}{8^{8}}

Simplify Exponents: Simplify the expression using the properties of exponents. $\newline$ We have the expression $(8^{2}\cdot5^{-4}\cdot8^{5})/(8^{8})$ . We can combine the powers of $8$ by adding the exponents in the numerator and then subtracting the exponent in the denominator.
Add Exponents: Add the exponents of $8$ in the numerator. $\newline$ $8^{2} \times 8^{5} = 8^{2+5} = 8^{7}.$
Divide Powers: Divide the powers of $8$ by subtracting the exponents. $\newline$ $(8^{7})/(8^{8}) = 8^{7-8} = 8^{-1}$ .
Simplify Power: Simplify the power of $5$ in the numerator. $\newline$ Since $5^{-4}$ is in the numerator, it is equivalent to $\frac{1}{5^{4}}$ .
Combine Powers: Combine the simplified powers of $8$ and $5$ . We have $8^{-1} \times \frac{1}{5^{4}}$ . Since $8^{-1}$ is the same as $\frac{1}{8}$ , we can rewrite the expression as $\left(\frac{1}{8}\right) \times \left(\frac{1}{5^{4}}\right)$ .
Multiply Fractions: Multiply the fractions. $\newline$ $(\frac{1}{8}) \times (\frac{1}{5^{4}}) = \frac{1}{8\times5^{4}}$ .
Calculate Denominator: Calculate the denominator. $\newline$ $8 \times 5^{4} = 8 \times (5\times5\times5\times5) = 8 \times 625 = 5000$ .
Final Expression: Write the final simplified expression. $\newline$ The expression simplifies to $\frac{1}{5000}$ .