Express all terms with a common base, which is 2.
This chemistry question involves key chemical concepts and calculations. The detailed solution below walks through each step, from identifying the reaction type to computing the final answer.
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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.
$$\log N = 0.8885 + \bar{3}.6628 - \bar{1}.8
