Question

16^1/5 * 2^x = 8^3/4. Solve for x using prime factorisation.​

Answer 1

`x=(29)/(20)`

Explanation:

`16^{(1)/(5)} \cdot 2^x=8^{(3)/(4)}`

Prime factorisation of 16:

16 ÷ 2 = 8

8 ÷ 2 = 4

4 ÷ 2 = 2

⇒ 16 = 2⁴

Prime factorisation of 8:

8 ÷ 2 = 4

4 ÷ 2 = 2

⇒ 8 = 2³

Substitute 16 for 2⁴ and 8 for 2³:

`\begin{aligned}16^{(1)/(5)} \cdot 2^x &=8^{(3)/(4)}\\\implies (2^4)^{(1)/(5)} \cdot 2^x &=(2^3)^{(3)/(4)}\end{aligned}`

`\textsf{Apply exponent rule} \quad (a^b)^c=a^(bc):`

`\implies 2^{(4)/(5)} \cdot 2^x=2^{(9)/(4)}`

`\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^(b+c):`

`\implies 2^{(4)/(5)+x}=2^{(9)/(4)}`

`\textsf{Apply exponent rule} \quad a^b=a^c \implies b=c:`

`\implies (4)/(5)+x=(9)/(4)`

Subject 4/5 from both sides:

`\implies x=(29)/(20)`