Express all terms with a common base, which is 2.

Okay, Big, let's tackle these problems.

Question 1: Solve for $x$: $4^{x+1} \times 2^{2x-3} = 512$

Step 1: Express all terms with a common base, which is 2.

$$4 = 2^2$$ $$512 = 2^9$$

Step 2: Substitute these into the equation.

$$(2^2)^{x+1} \times 2^{2x-3} = 2^9$$

Step 3: Apply the power rule $(a^m)^n = a^{mn}$.

$$2^{2(x+1)} \times 2^{2x-3} = 2^9$$ $$2^{2x+2} \times 2^{2x-3} = 2^9$$

Step 4: Apply the product rule $a^m \times a^n = a^{m+n}$.

$$2^{(2x+2) + (2x-3)} = 2^9$$ $$2^{4x-1} = 2^9$$

Step 5: Equate the exponents since the bases are the same.

$$4x-1 = 9$$

Step 6: Solve for $x$.

$$4x = 9+1$$ $$4x = 10$$ $$x = \frac{10}{4}$$ $$x = 2.5$$

The value of $x$ is $\boxed{2.5}$.

Question 2: Use logarithms table to evaluate: $\frac{7.735 \times 0.0046}{0.693}$

Step 1: Let $N = \frac{7.735 \times 0.0046}{0.693}$. Take the logarithm of both sides.

$$\log N = \log(7.735) + \log(0.0046) - \log(0.693)$$

Step 2: Find the logarithms of each number using logarithm tables.

$$\log(7.735) = 0.8885$$ $$\log(0.0046) = \bar{3}.6628$$ $$\log(0.693) = \bar{1}.8407$$

Step 3: Perform the addition and subtraction of the logarithms.

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