If \(x = a^3\) and \(y = a^7\), where \(a \neq 0\), which of the following is equivalent to \(a^{13}\)?
Show Hint
- \(xy\)
- \(x^2y\)
- $\dfrac{x^3}{y}$
- $\dfrac{x^4}{y}$
- $\dfrac{y^3}{x}$
The Correct Option is B
Solution and Explanation
This problem tests the rules of exponents, specifically the product rule (\(a^m \cdot a^n = a^{m+n}\)) and the power rule (\((a^m)^n = a^{mn}\)). We need to combine the given expressions for \(x\) and \(y\) to form the target expression \(a^{13}\).
Step 2: Key Formula or Approach:
We will test the given options by substituting the definitions of \(x\) and \(y\).
Given:
\(x = a^3\)
\(y = a^7\)
Step 3: Detailed Explanation:
Let's evaluate the expression in option (B), \(x^2y\).
1. First, substitute \(x = a^3\) into the expression:
\[ x^2y = (a^3)^2 y \] 2. Apply the power rule of exponents, \((a^m)^n = a^{mn}\), to simplify \((a^3)^2\):
\[ (a^3)^2 = a^{3 \times 2} = a^6 \] 3. The expression now becomes:
\[ a^6 y \] 4. Now, substitute \(y = a^7\) into this expression:
\[ a^6 \cdot a^7 \] 5. Apply the product rule of exponents, \(a^m \cdot a^n = a^{m+n}\), to combine the terms:
\[ a^{6 + 7} = a^{13} \] This matches the target expression.
For completeness, let's check option (A):
\[ xy = (a^3)(a^7) = a^{3+7} = a^{10} \] This is not equal to \(a^{13}\).
Step 4: Final Answer:
The expression equivalent to \(a^{13}\) is \(x^2y\).
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