Question

If 4^x - 4^x-1 = 24, then evaluate (2x)^x.
Plzz help me solve this

Answer 1

${4}^{x} - {4}^{x - 1} = 24 \\ \\ {4}^{x} - {4}^{x} \times {4}^{ - 1} = 24 \\ \\ {4}^{x} (1 - {4}^{ - 1}) = 24 \\ \\ {4}^{x}(1 - \frac{1}{4} ) = 24 \\ \\ {4}^{x} ( \frac{3}{4} ) = 24 \\ \\ {4}^{x} = \frac{24}{ \frac{3}{4} } \\ \\ {4}^{x} = \frac{96}{3} \\ \\ {4}^{x} = 32 \\ \\ { ({2}^{2} )}^{x} = {2}^{5} \\ \\ {2}^{2x} = {2}^{5} \\ \\ 2x = 5 \\ x = 2.5$

Now,

${2x}^{x} \\ \\ {(2 \times 2.5)}^{2.5} \\ \\ {5}^{2.5} \\ \\ {5}^{ \frac{25}{10} } = {5}^{ \frac{5}{2} } = {( {5}^{5} )}^{ \frac{1}{2} } = {3125}^{ \frac{1}{2} } = \sqrt{3125} = 25 \sqrt{5}$