Question

4(x+2)=\frac{1}{32}

Explanation:

Step1: Rewrite 4 and 32 as powers of 2

We know that \(4 = 2^2\) and \(32 = 2^5\), so the equation \(4^{(x + 2)}=\frac{1}{32}\) can be rewritten as \((2^2)^{(x + 2)}=\frac{1}{2^5}\).

Step2: Simplify the exponents

Using the power - of - a - power rule \((a^m)^n=a^{mn}\) on the left - hand side and the negative exponent rule \(a^{-n}=\frac{1}{a^n}\) on the right - hand side, we get \(2^{2(x + 2)}=2^{-5}\).

Step3: Set the exponents equal

Since the bases are the same (\(a^m=a^n\) implies \(m = n\) for \(a>0,a
eq1\)), we can set the exponents equal to each other: \(2(x + 2)=-5\).

Step4: Solve for x

First, expand the left - hand side: \(2x+4=-5\).
Then, subtract 4 from both sides: \(2x=-5 - 4=-9\).
Finally, divide both sides by 2: \(x=-\frac{9}{2}\).

Answer:

\(x = -\frac{9}{2}\)