Question

find or estimate the value of each variable mentally.
$4^a = 16$
$4^b = 2$
$4^{\frac{1}{2}} = c$
$4^d = 56$

Explanation:

Step1: Solve for $a$: match powers of 4

$4^a = 16 = 4^2$, so $a=2$

Step2: Solve for $b$: rewrite 2 as root of 4

$4^b = 2 = 4^{\frac{1}{2}}$, so $b=\frac{1}{2}$

Step3: Solve for $c$: compute fractional exponent

$4^{\frac{1}{2}} = \sqrt{4} = 2$, so $c=2$

Step4: Estimate $d$: use known powers of 4

$4^3=64$, $4^2=16$, so $56$ is between $4^2$ and $4^3$. $56\approx64\cdot0.875$, so $d\approx2.8$ (since $4^{2.8}\approx56$)

Answer:

$a=2$, $b=\frac{1}{2}$, $c=2$, $d\approx2.8$