Rewriting numbers as powers of 2

Text

Rewrite every number as a power of base $2$: $4^{\sqrt{2}};\quad 8^{1/2};\quad 3^{\pi};\quad \left(\tfrac{1}{2}\right)^{-3};\quad 16^{1/3};\quad 5^{e}.$

Solution

Write each base as a power of $2$; for $3$ and $5$ use $x=2^{\log_2 x}$.

• $4^{\sqrt2}=2^{2\sqrt2}$, exponent $\approx2{,}83$;
• $8^{1/2}=2^{3/2}$;
• $3^{\pi}=2^{\pi\log_2 3}$, exponent $\approx4{,}98$;
• $\left(\tfrac12\right)^{-3}=2^{3}$;
• $16^{1/3}=2^{4/3}$;
• $5^{e}=2^{e\log_2 5}$, exponent $\approx6{,}31$.

$\boxed{2^{2\sqrt2},\ 2^{3/2},\ 2^{\pi\log_2 3},\ 2^{3},\ 2^{4/3},\ 2^{e\log_2 5}}$