Question

1. Find the valu of x
$\large\sf {8}^{2x + 2} = {16}^{3x - 1}$
2. Evaluate :-
$\large\sf\frac{36^{ \frac{5}{2}} - 36^{ \frac{7}{2} } }{36^{ \frac{3}{2} } }$
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Answer 1

See the solution.

Hope it helps!!

Answer 2

1. As per the information provided in the question, We have :

• 8^(2x + 2) = 16^(3x - 1)

We are asked to find the value of x, In order to find the value of x we will simplify it using laws of exponents.

$\begin{gathered} \longmapsto \rm 8^{(2x + 2) }= 16^{(3x - 1)} \end{gathered}$

Let's make the bases equal, So that powers can be equated,

$\begin{gathered} \longmapsto \rm {2}^{{3}{(2x + 2) }}= {2}^{{4} {(3x - 1})} \end{gathered}$

$\begin{gathered} \longmapsto \rm {{3}{(2x + 2) }}= {{4} {(3x - 1})} \end{gathered}$

$\begin{gathered} \longmapsto \rm 6x + 6= 12x - 4 \end{gathered}$

$\begin{gathered} \longmapsto \rm 6x - 12x= - 4 - 6 \end{gathered}$

$\begin{gathered} \longmapsto \rm - 6x= - 10 \end{gathered}$

$\begin{gathered} \longmapsto \rm x= \cancel{ \dfrac{ - 10}{ - 6}} \end{gathered}$

$\begin{gathered} \longmapsto \rm x= \dfrac{ 5}{ 3} \end{gathered}$

∴ The value of x is 5/3.

$\rule{200}2$

2. As per the information provided in the question, We have :

• $\large\sf\dfrac{36^{ \frac{5}{2}} - 36^{ \frac{7}{2} } }{36^{ \frac{3}{2} } }$

We are asked to evaluate. In order to do it, We will use laws of exponents.

$\longmapsto \sf\dfrac{7776- 279936 }{ {2}^{3} \times {3}^{3} }$

$\longmapsto \sf\dfrac{ - 272160}{ {2}^{3} \times {3}^{3} }$

$\longmapsto \sf - \dfrac{ {2}^{5} \times {3}^{5} \times 5 \times 7 }{ {2}^{3} \times {3}^{3} }$

$\longmapsto \sf - \dfrac{ \cancel{{2}^{5}} \times \cancel{{3}^{5} } \times 5 \times 7}{ \cancel{{2}^{3} }\times \cancel{{3}^{3} } }$

$\longmapsto \sf - {{2}^{2} \times {3}^{2} \times 5 \times 7}$

$\longmapsto \sf -{ 4 \times 9 \times 35}$

$\longmapsto \sf -{1260}$

$\rule{200}2$