Assertion: `((-10)/(3))xx((-10)/(3))xx((-10)/(3))xx((-10)/(3))xx((-10)/(3))xx((-10)/(3))xx((-10)/(3))xx((-10)/(3))xx((-10)/(3))xx((-10)/(3))=((-10)^(10))/((3)^(10))`
Reason : Any number a can be written as, `a=m xx 10^n` in standard from, where m lies between 1 and 10 (including 1) and n is any integer.

Options

A. If both assertion and reason are true and reason is the corret explanation of assertion.
B. If both assertion and reason are true but reason is not the correct explanation of assertion.
C. If assertion is true but reason is false.
D. If assertion is false but reason is true.

Answer

To solve the assertion and reason problem step by step, we will analyze both the assertion and the reason provided.

### Step 1: Analyze the Assertion
The assertion states:
\[ \left(-\frac{10}{3}\right) \times \left(-\frac{10}{3}\right) \times \left(-\frac{10}{3}\right) \times \left(-\frac{10}{3}\right) \times \left(-\frac{10}{3}\right) \times \left(-\frac{10}{3}\right) \times \left(-\frac{10}{3}\right) \times \left(-\frac{10}{3}\right) \times \left(-\frac{10}{3}\right) \times \left(-\frac{10}{3}\right) = \frac{(-10)^{10}}{3^{10}} \]

- We have \(-\frac{10}{3}\) multiplied by itself 10 times.
- According to the laws of exponents, multiplying the same base \(a\) \(n\) times can be expressed as \(a^n\).
- Here, our base is \(-\frac{10}{3}\) and we multiply it 10 times.

Thus, we can rewrite the left-hand side as:
\[ \left(-\frac{10}{3}\right)^{10} \]

### Step 2: Simplify the Left-Hand Side
Now, we simplify \(\left(-\frac{10}{3}\right)^{10}\):
\[ \left(-\frac{10}{3}\right)^{10} = \frac{(-10)^{10}}{3^{10}} \]

### Step 3: Conclusion of Assertion
This matches the right-hand side of the assertion:
\[ \frac{(-10)^{10}}{3^{10}} \]
Thus, the assertion is **true**.

### Step 4: Analyze the Reason
The reason states:
"Any number \(a\) can be written as \(a = m \times 10^n\) in standard form, where \(m\) lies between 1 and 10 (including 1) and \(n\) is any integer."

- This statement is indeed true. For example, \(20\) can be expressed as \(2 \times 10^1\), and \(0.0023\) can be expressed as \(2.3 \times 10^{-3}\).
- The condition that \(m\) must lie between 1 and 10 and \(n\) is an integer is also correct.

### Step 5: Conclusion of Reason
The reason is **true**, but it does not explain the assertion directly. The assertion is about the multiplication of a fraction, while the reason discusses the representation of numbers in standard form.

### Final Conclusion
Both the assertion and the reason are true, but the reason does not provide a correct explanation for the assertion. Therefore, the correct option is:
- Assertion and reason are true, but the reason is not the correct explanation of the assertion.