7. (a) \( 3^{x}=\sqrt[4]{3^{5}} \) Find the value of \( x \). \( \square \) :mwilecth givinu \( \square \) -hubloin bluedz uoy (b) Simplify \( \left(32 y^{15}\right)^{\frac{2}{2}} \). 8. Find the values of \( m \) and \( n \). (a) \( 2^{m}=0.125 \) Anstiots ad -104. fais inuanm pa patini 6 al nothamolal gmtos insingho iil gized smae griad tamal bidssingosen -ationh mionalias 10 - inje smoz Aifive noitacmetm litahupe sit (b) \( 2^{4 n} \times 2^{2 n}=512 \)

Solve the equation by following steps: - step0: Solve for \(m\): \(2^{m}=0.125\) - step1: Rewrite the expression: \(2^{m}=\frac{1}{8}\) - step2: Rewrite the expression: \(2^{m}=2^{-3}\) - step3: Set the exponents equal: \(m=-3\) Calculate or simplify the expression \( (32 y^{15})^{2/2} \). Simplify the expression by following steps: - step0: Solution: \(\left(32y^{15}\right)^{\frac{2}{2}}\) - step1: Divide the terms: \(\left(32y^{15}\right)^{1}\) - step2: Calculate: \(32y^{15}\) Solve the equation \( 2^{4n} \times 2^{2n}=512 \). Solve the equation by following steps: - step0: Solve for \(n\): \(2^{4n}\times 2^{2n}=512\) - step1: Multiply the terms: \(2^{6n}=512\) - step2: Rewrite in exponential form: \(2^{6n}=2^{9}\) - step3: Set the exponents equal: \(6n=9\) - step4: Divide both sides: \(\frac{6n}{6}=\frac{9}{6}\) - step5: Divide the numbers: \(n=\frac{3}{2}\) Solve the equation \( 3^{x}=3^{5/4} \). Solve the equation by following steps: - step0: Solve for \(x\): \(3^{x}=3^{\frac{5}{4}}\) - step1: Set the exponents equal: \(x=\frac{5}{4}\) Let's solve the problems step by step. ### Problem 7 #### (a) Solve \( 3^{x}=\sqrt[4]{3^{5}} \) To solve this equation, we can rewrite the right side: \[ \sqrt[4]{3^{5}} = 3^{5/4} \] Now we have: \[ 3^{x} = 3^{5/4} \] Since the bases are the same, we can set the exponents equal to each other: \[ x = \frac{5}{4} \] Thus, the value of \( x \) is: \[ x = 1.25 \] #### (b) Simplify \( \left(32 y^{15}\right)^{\frac{2}{2}} \) The expression simplifies as follows: \[ \left(32 y^{15}\right)^{\frac{2}{2}} = 32 y^{15} \] So, the simplified form is: \[ 32 y^{15} \] --- ### Problem 8 #### (a) Solve \( 2^{m}=0.125 \) We can express \( 0.125 \) as a power of \( 2 \): \[ 0.125 = \frac{1}{8} = 2^{-3} \] Thus, we have: \[ 2^{m} = 2^{-3} \] Setting the exponents equal gives us: \[ m = -3 \] #### (b) Solve \( 2^{4n} \times 2^{2n}=512 \) We can combine the left side: \[ 2^{4n + 2n} = 2^{6n} \] Now, we know that \( 512 = 2^9 \), so we can set the exponents equal: \[ 6n = 9 \] Solving for \( n \): \[ n = \frac{9}{6} = \frac{3}{2} = 1.5 \] ### Summary of Results - For \( 7(a) \), \( x = 1.25 \) - For \( 7(b) \), the simplified expression is \( 32 y^{15} \) - For \( 8(a) \), \( m = -3 \) - For \( 8(b) \), \( n = 1.5 \)