Question

find or estimate the value of each variable mentally.
$4^a = 16$
$4^b = 2$
$4^{\frac{3}{2}} = c$
$4^d = 56$

Explanation:

Step1: Solve for \(a\): match powers of 4

We know \(4^2 = 16\), so \(a = 2\).

Step2: Solve for \(b\): rewrite 2 as root of 4

Since \(4^{1/2} = \sqrt{4} = 2\), so \(b = \frac{1}{2}\).

Step3: Solve for \(c\): compute fractional power

\(4^{\frac{3}{2}} = (4^{1/2})^3 = 2^3 = 8\), so \(c = 8\).

Step4: Estimate \(d\): compare powers of 4

We know \(4^3 = 64\) and \(4^2 = 16\). 56 is between 16 and 64. Since \(56 = 7 \times 8 = 7 \times 4^{1.5}\), or we use logarithms: \(d = \log_4 56 = \frac{\ln 56}{\ln 4} \approx \frac{4.025}{1.386} \approx 2.90\).

Answer:

\(a = 2\), \(b = \frac{1}{2}\), \(c = 8\), \(d \approx 2.90\)