25. (a) \( (\sqrt{a})^{7}=k \sqrt{a} \), where \( k=a^{\prime} \) Find the value of \( n \).

To solve this problem, we need to simplify the left side of the equation and then compare the exponents to find the value of \( n \). Given: \[ (\sqrt{a})^{7} = k \sqrt{a} \] where \( k = a^{n} \). First, let's simplify the left side of the equation. Since \( \sqrt{a} \) is the same as \( a^{1/2} \), we can rewrite the left side as: \[ (a^{1/2})^{7} \] Now, when we raise a power to a power, we multiply the exponents: \[ a^{(1/2) \cdot 7} = a^{7/2} \] So the equation becomes: \[ a^{7/2} = k \sqrt{a} \] Since \( k = a^{n} \) and \( \sqrt{a} = a^{1/2} \), we can rewrite the equation as: \[ a^{7/2} = a^{n} \cdot a^{1/2} \] Now, we can combine the exponents on the right side of the equation using the rule \( a^m \cdot a^n = a^{m+n} \): \[ a^{7/2} = a^{n + 1/2} \] Since the bases are the same, we can equate the exponents: \[ 7/2 = n + 1/2 \] To solve for \( n \), we subtract \( 1/2 \) from both sides of the equation: \[ 7/2 - 1/2 = n \] \[ 6/2 = n \] \[ n = 3 \] So the value of \( n \) is 3.