Find the value of diamond : ((5^(2))/(5^(diamond)))^(3)=5^(24)

Identify Equation & Property: Identify the given equation and the property of exponents that when dividing like bases, we subtract the exponents. The equation is ((5^(2))/(5^(diamond)))^(3)=5^(24). Using the property of exponents, we get (5^(2 - diamond))^3 = 5^(24).Apply Power to Exponent: Simplify the equation by applying the power to the exponent inside the parentheses. (5^(2 - diamond))^3 = 5^(3*(2 - diamond)) = 5^(24).Set Exponents Equal: Since the bases are the same, we can set the exponents equal to each other. 3*(2 - diamond) = 24.Divide to Solve: Divide both sides of the equation by 3 to solve for the variable ＂diamond＂. (3*(2 - diamond))/3 = 24/3.Isolate & Simplify: Simplify the equation to isolate the term with ＂diamond＂. 2 - diamond = 8.Add & Subtract to Solve: Add ＂diamond＂ to both sides and subtract 8 from both sides to solve for ＂diamond＂. diamond = 2 - 8.Calculate Final Value: Calculate the final value of ＂diamond＂. diamond = -6.

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Q. Find the value of

\diamond

\newline

\left(\frac{5^{2}}{5^{\diamond}}\right)^{3}=5^{24}

Identify Equation & Property: Identify the given equation and the property of exponents that when dividing like bases, we subtract the exponents. $\newline$ The equation is $(\frac{5^{2}}{5^{\diamond}})^{3}=5^{24}$ . $\newline$ Using the property of exponents, we get $(5^{2 - \diamond})^3 = 5^{24}$ .
Apply Power to Exponent: Simplify the equation by applying the power to the exponent inside the parentheses. $\newline$ $(5^{(2 - \text{diamond})})^3 = 5^{3*(2 - \text{diamond})} = 5^{24}$ .
Set Exponents Equal: Since the bases are the same, we can set the exponents equal to each other. $3\times(2 - \diamond) = 24$ .
Divide to Solve: Divide both sides of the equation by $3$ to solve for the variable ＂diamond＂. $\newline$ $\frac{3(2 - \text{diamond})}{3} = \frac{24}{3}.$
Isolate & Simplify: Simplify the equation to isolate the term with ＂diamond＂. $\newline$ $2 - \text{diamond} = 8$ .
Add & Subtract to Solve: Add ＂diamond＂ to both sides and subtract $8$ from both sides to solve for ＂diamond＂. $\newline$ $\text{diamond} = 2 - 8.$
Calculate Final Value: Calculate the final value of ＂diamond＂. $\newline$ $diamond = -6$ .