Simplify. (5^(-x)×25^(2y))/(125^(-(x)/(3)))

Express Bases in Terms of 5: We start by expressing all the bases in terms of the prime number 5, since 25 and 125 are powers of 5. 25 is 5 squared, so 25^(2y) = (5^2)^(2y). 125 is 5 cubed, so 125^(-(x)/(3)) = (5^3)^(-(x)/(3)).Apply Power of a Power Rule: Now we apply the power of a power rule, which states that (a^b)^c = a^(b*c). For 25^(2y), we have (5^2)^(2y) = 5^(2*2y) = 5^(4y). For 125^(-(x)/(3)), we have (5^3)^(-(x)/(3)) = 5^(3*(-(x)/(3))) = 5^(-x).Rewrite Using Simplifications: We can now rewrite the original expression using these simplifications: (5^(-x)×25^(2y))/(125^(-(x)/(3))) = (5^(-x)×5^(4y))/(5^(-x)).Apply Multiplication Rule: Next, we apply the multiplication rule for exponents with the same base, which states that a^b * a^c = a^(b+c). So, 5^(-x)×5^(4y) = 5^(-x+4y).Apply Division Rule: Now we have the expression (5^(-x+4y))/(5^(-x)). We apply the division rule for exponents with the same base, which states that a^b / a^c = a^(b-c). So, 5^(-x+4y) / 5^(-x) = 5^((-x+4y)-(-x)) = 5^(4y+x).Final Simplified Expression: We have simplified the expression to 5^(4y+x). This is the final form of the expression, and there are no further simplifications to be made.

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Simplify.
(5^(-x)×25^(2y))/(125^(-(x)/(3)))

Simplify. $\newline$ $\frac{5^{-x} \times 25^{2 y}}{125^{-\frac{x}{3}}}$

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Algebra 1

Evaluate integers raised to rational exponents

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Q. Simplify.

\newline

\frac{5^{-x} \times 25^{2 y}}{125^{-\frac{x}{3}}}

Express Bases in Terms of $5$ : We start by expressing all the bases in terms of the prime number $5$ , since $25$ and $125$ are powers of $5$ . $\newline$ $25$ is $5$ squared, so $25^{2y} = (5^2)^{2y}$ . $\newline$ $125$ is $5$ cubed, so $125^{-(x)/(3)} = (5^3)^{-(x)/(3)}$ .
Apply Power of a Power Rule: Now we apply the power of a power rule, which states that a^b)^c = a^{(b*c)}\. For \$25^{(2y)}, we have 5^2)^{(2y)} = 5^{(2*2y)} = 5^{(4y)}\. For \$125^{-(x)/(3)}, we have \(5^ $3$ )^{-(x)/( $3$ )} = $5$ ^{ $3$ *(-(x)/( $3$ ))} = $5$ ^{-x}\.
Rewrite Using Simplifications: We can now rewrite the original expression using these simplifications: $\newline$ $(5^{-x}\times25^{2y})/(125^{-\frac{x}{3}}) = (5^{-x}\times5^{4y})/(5^{-x})$ .
Apply Multiplication Rule: Next, we apply the multiplication rule for exponents with the same base, which states that $a^b \times a^c = a^{(b+c)}$ . $\newline$ So, $5^{-x}\times5^{4y} = 5^{(-x+4y)}$ .
Apply Division Rule: Now we have the expression $(5^{-x+4y})/(5^{-x})$ . We apply the division rule for exponents with the same base, which states that $a^b / a^c = a^{b-c}$ . So, $5^{-x+4y} / 5^{-x} = 5^{(-x+4y)-(-x)} = 5^{4y+x}$ .
Final Simplified Expression: We have simplified the expression to $5^{4y+x}$ . This is the final form of the expression, and there are no further simplifications to be made.