Answer:
[tex]x=\dfrac{29}{20}[/tex]
Step-by-step explanation:
[tex]16^{\frac{1}{5}} \cdot 2^x=8^{\frac{3}{4}}[/tex]
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:
[tex]\begin{aligned}16^{\frac{1}{5}} \cdot 2^x &=8^{\frac{3}{4}}\\\implies (2^4)^{\frac{1}{5}} \cdot 2^x &=(2^3)^{\frac{3}{4}}\end{aligned}[/tex]
[tex]\textsf{Apply exponent rule} \quad (a^b)^c=a^{bc}:[/tex]
[tex]\implies 2^{\frac{4}{5}} \cdot 2^x=2^{\frac{9}{4}}[/tex]
[tex]\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^{b+c}:[/tex]
[tex]\implies 2^{\frac{4}{5}+x}=2^{\frac{9}{4}}[/tex]
[tex]\textsf{Apply exponent rule} \quad a^b=a^c \implies b=c:[/tex]
[tex]\implies \dfrac{4}{5}+x=\dfrac{9}{4}[/tex]
Subject 4/5 from both sides:
[tex]\implies x=\dfrac{29}{20}[/tex]
