- Use the power of a product rule \((ab)^n = a^n b^n\) to distribute the exponent.
- Apply the power of a power rule \((x^m)^n = x^{m \cdot n}\) to the variable term.
- Calculate the constant square: \(5^2 = 25\).
- Multiply the results to arrive at the simplified expression: \(\boxed{25k^{12}}\).
Explanation
- Analyzing the Expression
To simplify the expression \((5k^6)^2\), we identify the base as a product of a constant and a variable raised to a power. The expression is of the form \((ab)^n\), where \(a = 5\), \(b = k^6\), and \(n = 2\). We can utilize the power of a product rule, \((ab)^n = a^n b^n\), to distribute the exponent to each factor within the parentheses.
- Applying Exponent Laws
Applying the power of a product rule, we have:
\[(5k^6)^2 = 5^2 \cdot (k^6)^2\]
Now we calculate the constant factor:
\[5^2 = 25\]
Next, we simplify the variable part using the power of a power rule, which states that \((x^m)^n = x^{m \cdot n}\). Applying this to \((k^6)^2\):
\[(k^6)^2 = k^{6 \cdot 2} = k^{12}\]
- Final Simplification
By combining the simplified constant and variable parts, we obtain the final result:
\[25 \cdot k^{12} = 25k^{12}\]
Since the exponent 12 is positive, our simplification is complete as requested. Thus, the simplified expression is \(\boxed{25k^{12}}\).
Examples
In physics, when calculating the area of a square-shaped solar panel where each side is given by \(5k^6\) meters, the area would be found by squaring the side length, \((5k^6)^2 = 25k^{12}\) square meters. This demonstrates how exponent rules help simplify calculations in geometric and scientific applications.