Question

1. $16^{r} cdot 64^{3 - 3r} = 64$

Explanation:

Step1: Express bases as powers of 4

Since \(16 = 4^2\) and \(64 = 4^3\), we rewrite the equation:
\((4^2)^r \cdot (4^3)^{3 - 3r} = 4^3\)

Step2: Simplify exponents using power rule

Using \((a^m)^n = a^{mn}\), we get:
\(4^{2r} \cdot 4^{9 - 9r} = 4^3\)

Step3: Combine exponents using product rule

Using \(a^m \cdot a^n = a^{m + n}\), we have:
\(4^{2r + 9 - 9r} = 4^3\)
Simplify the exponent:
\(4^{9 - 7r} = 4^3\)

Step4: Set exponents equal (since bases are equal)

\(9 - 7r = 3\)

Step5: Solve for r

Subtract 9 from both sides:
\(-7r = 3 - 9\)
\(-7r = -6\)
Divide both sides by -7:
\(r = \frac{6}{7}\)

Answer:

\(r = \frac{6}{7}\)