Question

47. Assertion: If (16) ^ (2x + 3) = (64) ^ (x + 3) then 4 ^ (2x - 3) = 64 .

Reason: If a ne0, pm1 then a ^ m = a ^ n m = n (a ^ m) ^ n = a ^ (mn)

Answer 1

To analyze the assertion and reason provided, we first need to break down the given equation:

1. **Assertion:** If \( (16)^{(2x + 3)} = (64)^{(x + 3)} \), then show that \( 4^{(2x - 3)} = 64 \).

2. **Reason:** If \( a \neq 0 \) and \( a \neq 1 \), then \( a^m = a^n \) implies \( m = n \), and \( (a^m)^n = a^{(mn)} \).

### Step 1: Analyze the Assertion

We start with the equation:
\[
(16)^{(2x + 3)} = (64)^{(x + 3)}
\]

Both 16 and 64 can be expressed in terms of base 4:
\[
16 = 4^2 \quad \text{and} \quad 64 = 4^3
\]

Now, substituting these in the equation gives:
\[
(4^2)^{(2x + 3)} = (4^3)^{(x + 3)}
\]

Using the power of a power property \( (a^m)^n = a^{mn} \):
\[
4^{(2(2x + 3))} = 4^{(3(x + 3))}
\]

This simplifies to:
\[
4^{(4x + 6)} = 4^{(3x + 9)}
\]

Since the bases are the same, we can equate the exponents:
\[
4x + 6 = 3x + 9
\]
Subtract \( 3x \) from both sides:
\[
x + 6 = 9
\]
Now, isolate \( x \):
\[
x = 3
\]

### Step 2: Verify the Result in the Assertion

Now, we need to check if \( 4^{(2x - 3)} = 64 \):
\[
2x - 3 = 2(3) - 3 = 6 - 3 = 3
\]

Now we compute \( 4^3 \):
\[
4^3 = 64
\]

Indeed, \( 4^{(2x - 3)} = 64 \).

### Step 3: Analyze the Reason

The reason states the property of exponents that \( a^m = a^n \) implies \( m = n \) when \( a \neq 0 \) and \( a \neq 1 \). This property is correctly applied in deducing the equality of exponents.

### Conclusion

Since we proved the assertion true and validated the reasoning behind it, we can conclude:

- The **assertion** is true.
- The **reason** is a valid justification for the assertion.

Therefore, both the assertion and the reason are correct. The final conclusion is:

Both the assertion and the reason are true.