What is value of `(a^(10))^(3) / (a^(6))^(5) `?

Options

A. a
B. `a^(2)`
C. 1
D. 1/a

Answer

To solve the expression \((a^{10})^{3} / (a^{6})^{5}\), we will follow the rules of exponents step by step.

### Step-by-Step Solution:

1. **Apply the Power of a Power Rule**:
According to the exponent rule \( (a^m)^n = a^{m \cdot n} \), we can simplify both the numerator and the denominator.
\[ (a^{10})^{3} = a^{10 \cdot 3} = a^{30} \]
\[ (a^{6})^{5} = a^{6 \cdot 5} = a^{30} \]

2. **Rewrite the Expression**:
Now we can rewrite the original expression using the results from step 1:
\[ \frac{(a^{10})^{3}}{(a^{6})^{5}} = \frac{a^{30}}{a^{30}} \]

3. **Apply the Quotient Rule**:
According to the exponent rule \( \frac{a^m}{a^n} = a^{m-n} \), we can simplify the fraction:
\[ \frac{a^{30}}{a^{30}} = a^{30 - 30} = a^{0} \]

4. **Evaluate \( a^{0} \)**:
By the property of exponents, any non-zero number raised to the power of 0 is equal to 1:
\[ a^{0} = 1 \]

### Final Answer:
Thus, the value of \((a^{10})^{3} / (a^{6})^{5}\) is \(1\).

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