Question

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Answer 1

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Answer 2

Question :-

If 4^(x + 1) - 4^x = 24, then find the value of (2x)^x is :

(a) 2√3

(b) 3√3

(c) 4√3

(d) √3

Answer :-

Value of (2x)^x is (b) 3√3

Explanation :-

$\mathsf{ 4^{x + 1} - 4^x = 24 } \\ \\ \\ \implies \mathsf{ 4^x( 4^{1}) - 4^x = 24 }$

$\boxed{\boxed{ \bf \because {a}^{m + n} = {a}^{m} \times {a}^{n} }}$

$\mathsf{ \implies4^x( 4) - 4^x = 24 }$

Taking 4^x common

$\mathsf{ \implies4^x( 4 - 1) = 24 }$

$\mathsf{ \implies4^x(3) = 24 }$

$\mathsf{ \implies4^x= \dfrac{24}{3} }$

$\mathsf{ \implies 4^x= 8 }$

$\mathsf{ \implies (2^2)^x= {2}^{3} }$

$\mathsf{ \implies 2^{2x}= {2}^{3} }$

$\boxed{\boxed{ \bf \because ({a}^{m} )^{n} = {a}^{mn}}}$

If bases are equal we can equate powers

$\mathsf{ \implies 2x = 3 }$

$\mathsf{ \implies x = \dfrac{3}{2} }$

Now, (2x)^x

$\mathsf{ ( 2x)^x}$

$\mathsf{ = [2( \dfrac{3}{2}) ]^{ \dfrac{3}{2} } }$

$\mathsf{ = 3^{ \dfrac{3}{2} } }$

Writing it in Radical form

$\mathsf{ = \sqrt{ {3}^{3} } }$

$\mathsf{ = \sqrt{27} }$

$\mathsf{ = \sqrt{9 \times 3} }$

$\mathsf{ = \sqrt{9} \times \sqrt{3} }$

$\mathsf{ = 3 \times \sqrt{3} }$

$\mathsf{ = 3\sqrt{3} }$

Hence, the value of (2x)^x is 3√3