Question

Simplify the expression answer should have a positive exponent six to the third power in parentheses times two the power of 2×6 to the -4 power over six to the second power

Answer 1

To simplify the expression \((6^3 \cdot 2^{2 \cdot 6^{-4}}) / (6^2)\), follow these steps:

1. **Combine the powers of \(6\) in the numerator**:
\[
6^3 \cdot 6^{-4} = 6^{3 + (-4)} = 6^{-1}
\]

2. **Re-write the expression**:
The expression now becomes:
\[
\frac{6^{-1} \cdot 2^{2}}{6^{2}}
\]

3. **Combine the \(6\) terms in the denominator**:
Now apply the property of exponents:
\[
\frac{6^{-1}}{6^{2}} = 6^{-1 - 2} = 6^{-3}
\]

4. **Putting it all together**:
We now have:
\[
\frac{2^2}{6^3}
\]

Calculate \(2^2 = 4\):
\[
= \frac{4}{6^3}
\]

5. **Translate \(6^3\)**:
We can express \(6^3\) as \(216\), but to keep the expression concise:
\[
= \frac{4}{6^3}
\]

6. **Final expression with positive exponents**:
To express it fully:
\[
= 4 \cdot 6^{-3}
\]

This implies, the final simplified expression is:
\[
\frac{4}{216} \quad \text{or} \quad 4 \cdot 6^{-3}
\]
Both forms are acceptable depending on your preference!