Question

1. $81 cdot 9^{-2b - 2} = 27$

Explanation:

Step1: Express all terms as powers of 3

We know that \(81 = 3^4\), \(9 = 3^2\), and \(27 = 3^3\). Substitute these into the equation:
\(3^4 \cdot (3^2)^{-2b - 2} = 3^3\)

Step2: Simplify the exponents

Using the power of a power rule \((a^m)^n = a^{mn}\), we get:
\(3^4 \cdot 3^{-4b - 4} = 3^3\)

Step3: Use the product of powers rule

The product of powers rule states that \(a^m \cdot a^n = a^{m + n}\). Applying this:
\(3^{4 + (-4b - 4)} = 3^3\)
Simplify the exponent:
\(3^{-4b} = 3^3\)

Step4: Set the exponents equal

Since the bases are the same and the equation holds, the exponents must be equal:
\(-4b = 3\)

Step5: Solve for \(b\)

Divide both sides by \(-4\):
\(b = -\frac{3}{4}\)

Answer:

\(b = -\frac{3}{4}\)