Question

Find the value of x in each of the following.

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

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

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

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

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

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\sqrt[5]{5x+2}=2 \\ \Rightarrow {\left(5x+2\right)}^{\frac{1}{5}}=2 \\ \Rightarrow {\left[{\left(5x+2\right)}^{\frac{1}{5}}\right]}^5={\left(2\right)}^5 \\ \Rightarrow \left(5x+2\right)=32 \\ \Rightarrow 5x=32-2 \\ \Rightarrow 5x=30 \\ \Rightarrow x=\frac{30}{5} \\ \Rightarrow x=6 \end{gathered}$$
Hence, the value of x is 6.

$$\begin{gathered} \left(\mathrm{ii}\right)\sqrt[3]{3x-2}=4 \\ \Rightarrow {\left(3x-2\right)}^{\frac{1}{3}}=4 \\ \Rightarrow {\left[{\left(3x-2\right)}^{\frac{1}{3}}\right]}^3={\left(4\right)}^3 \\ \Rightarrow \left(3x-2\right)=64 \\ \Rightarrow 3x=64+2 \\ \Rightarrow 3x=66 \\ \Rightarrow x=\frac{66}{3} \\ \Rightarrow x=22 \end{gathered}$$
Hence, the value of x is 22.

$$\begin{gathered} \left(\mathrm{iii}\right){\left(\frac{3}{4}\right)}^3{\left(\frac{4}{3}\right)}^{-7}={\left(\frac{3}{4}\right)}^{2x} \\ \Rightarrow {\left(\frac{3}{4}\right)}^3{\left(\frac{3}{4}\right)}^7={\left(\frac{3}{4}\right)}^{2x} \\ \Rightarrow {\left(\frac{3}{4}\right)}^{3+7}={\left(\frac{3}{4}\right)}^{2x} \\ \Rightarrow {\left(\frac{3}{4}\right)}^{10}={\left(\frac{3}{4}\right)}^{2x} \\ \Rightarrow 10=2x \\ \Rightarrow \frac{10}{2}=x \\ \Rightarrow 5=x \end{gathered}$$
Hence, the value of x is 5.

$$\begin{gathered} \left(\mathrm{iv}\right)5^{x-3}\times 3^{2x-8}=225 \\ \Rightarrow 5^{x-3}\times 3^{2x-8}={\left(15\right)}^2 \\ \Rightarrow 5^{x-3}\times 3^{2x-8}=5^2\times 3^2 \\ \Rightarrow x-3=2\mathrm{and}2x-8=2 \\ \Rightarrow x=2+3\mathrm{and}2x=2+8 \\ \Rightarrow x=5\mathrm{and}2x=10 \\ \Rightarrow x=5\mathrm{and}x=\frac{10}{2} \\ \Rightarrow x=5\mathrm{and}x=5 \\ \Rightarrow x=5 \end{gathered}$$
Hence, the value of x is 5.

$$\begin{gathered} \left(\mathrm{v}\right)\frac{3^{3x}·3^{2x}}{3^x}=\sqrt[4]{3^{20}} \\ \Rightarrow \frac{3^{3x+2x}}{3^x}={\left(3^{20}\right)}^{\frac{1}{4}} \\ \Rightarrow \frac{3^{5x}}{3^x}=3^5 \\ \Rightarrow 3^{5x-x}=3^5 \\ \Rightarrow 3^{4x}=3^5 \\ \Rightarrow 4x=5 \\ \Rightarrow x=\frac{5}{4} \end{gathered}$$
Hence, the value of x is $\frac{5}{4}$.