Question

1. $2^{x} cdot \frac{1}{32} = 32$

Explanation:

Step1: Multiply both sides by 32

To eliminate the fraction, we multiply both sides of the equation \(2^{x}\cdot\frac{1}{32}=32\) by 32. This gives us \(2^{x}\cdot\frac{1}{32}\times32 = 32\times32\). Simplifying the left side, the 32 in the numerator and denominator cancels out, so we have \(2^{x}=32\times32\).

Step2: Calculate the right side

We know that \(32 = 2^{5}\), so \(32\times32=2^{5}\times2^{5}\). Using the rule of exponents \(a^{m}\times a^{n}=a^{m + n}\), we get \(2^{5}\times2^{5}=2^{5 + 5}=2^{10}\). So now our equation is \(2^{x}=2^{10}\).

Step3: Solve for x

Since the bases are the same (both are 2) and the exponential function \(y = 2^{x}\) is one - to - one (if \(2^{a}=2^{b}\), then \(a = b\)), we can set the exponents equal to each other. So \(x = 10\).

Answer:

\(x = 10\)