# Find the value of x in each of the following.

Question:

Find the value of x in each of the following.

(i) $\sqrt[5]{5 x+2}=2$

(ii) $\sqrt[3]{3 x-2}=4$

(iii) $\left(\frac{3}{4}\right)^{3}\left(\frac{4}{3}\right)^{-7}=\left(\frac{3}{4}\right)^{2 x}$

(iv) $5^{x-3} \times 3^{2 x-8}=225$

(v) $\frac{3^{3 x} \cdot 3^{2 x}}{3^{x}}=\sqrt[4]{3^{20}}$

Solution:

(i) $\sqrt[5]{5 x+2}=2$

$\Rightarrow(5 x+2)^{\frac{1}{5}}=2$

$\Rightarrow\left[(5 x+2)^{\frac{1}{5}}\right]^{5}=(2)^{5}$

$\Rightarrow(5 x+2)=32$

$\Rightarrow 5 x=32-2$

$\Rightarrow 5 x=30$

$\Rightarrow x=\frac{30}{5}$

$\Rightarrow x=6$

Hence, the value of x is 6.

(ii) $\sqrt[3]{3 x-2}=4$

$\Rightarrow(3 x-2)^{\frac{1}{3}}=4$

$\Rightarrow\left[(3 x-2)^{\frac{1}{3}}\right]^{3}=(4)^{3}$

$\Rightarrow(3 x-2)=64$

$\Rightarrow 3 x=64+2$

$\Rightarrow 3 x=66$

$\Rightarrow x=\frac{66}{3}$

$\Rightarrow x=22$

Hence, the value of x is 22

(iii) $\left(\frac{3}{4}\right)^{3}\left(\frac{4}{3}\right)^{-7}=\left(\frac{3}{4}\right)^{2 x}$

$\Rightarrow\left(\frac{3}{4}\right)^{3}\left(\frac{3}{4}\right)^{7}=\left(\frac{3}{4}\right)^{2 x}$

$\Rightarrow\left(\frac{3}{4}\right)^{3+7}=\left(\frac{3}{4}\right)^{2 x}$

$\Rightarrow\left(\frac{3}{4}\right)^{10}=\left(\frac{3}{4}\right)^{2 x}$

$\Rightarrow 10=2 x$

$\Rightarrow \frac{10}{2}=x$

$\Rightarrow 5=x$

Hence, the value of x is 5.

(iv) $5^{x-3} \times 3^{2 x-8}=225$

$\Rightarrow 5^{x-3} \times 3^{2 x-8}=(15)^{2}$

$\Rightarrow 5^{x-3} \times 3^{2 x-8}=5^{2} \times 3^{2}$

$\Rightarrow x-3=2$ and $2 x-8=2$

$\Rightarrow x=2+3$ and $2 x=2+8$

$\Rightarrow x=5$ and $2 x=10$

$\Rightarrow x=5$ and $x=\frac{10}{2}$

$\Rightarrow x=5$ and $x=5$

$\Rightarrow x=5$

Hence, the value of x is 5.

(v) $\frac{3^{3 x} \cdot 3^{2 x}}{3^{x}}=\sqrt[4]{3^{20}}$

$\Rightarrow \frac{3^{3 x+2 x}}{3^{x}}=\left(3^{20}\right)^{\frac{1}{4}}$

$\Rightarrow \frac{3^{5 x}}{3^{x}}=3^{5}$

$\Rightarrow 3^{5 x-x}=3^{5}$

$\Rightarrow 3^{4 x}=3^{5}$

$\Rightarrow 4 x=5$

$\Rightarrow x=\frac{5}{4}$

Hence, the value of $x$ is $\frac{5}{4}$.