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

Simplify the Base: First, simplify the base of the equation using the properties of exponents. ((y^(5))/(y^(-3))) = y^(5 - (-3)) = y^(5 + 3) = y^8Apply Exponent Rule: Now, we have (y^8)^(diamond) = y^(24). According to the properties of exponents, when we raise a power to a power, we multiply the exponents. (y^8)^(diamond) = y^(8 * diamond)Set Up Equation: To find the value of diamond, we set the exponent of y on the left side of the equation equal to the exponent of y on the right side of the equation. 8 * diamond = 24Solve for Diamond: Divide both sides of the equation by 8 to solve for diamond. diamond = 24 / 8Calculate Final Value: Calculate the value of diamond. diamond = 3

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Q. Find the value of

\diamond

\newline

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

Simplify the Base: First, simplify the base of the equation using the properties of exponents. $\left(\frac{y^{5}}{y^{-3}}\right) = y^{5 - (-3)} = y^{5 + 3} = y^8$
Apply Exponent Rule: Now, we have $(y^8)^{\diamond} = y^{24}$ . According to the properties of exponents, when we raise a power to a power, we multiply the exponents. $\newline$ $(y^8)^{\diamond} = y^{8 \times \diamond}$
Set Up Equation: To find the value of diamond, we set the exponent of $y$ on the left side of the equation equal to the exponent of $y$ on the right side of the equation. $8 \times \text{diamond} = 24$
Solve for Diamond: Divide both sides of the equation by $8$ to solve for diamond. $\newline$ $\text{diamond} = \frac{24}{8}$
Calculate Final Value: Calculate the value of diamond. $\text{diamond} = 3$