The problem requires us to simplify four algebraic expressions. (a) $\frac{a^4 \times a^3}{a^2}$ (b) $\frac{(3x)^3 \times (4x^2)^2}{2x^4}$ (c) $\frac{e^n \times e^m}{e^p}$ (d) $\frac{6b(a^2b)^2}{t(2k)^2} \div \frac{3(ab)^2}{(2kt^2)^3}$

1. Problem Description

The problem requires us to simplify four algebraic expressions.

(a) $\frac{a^4 \times a^3}{a^2}$

(b) $\frac{(3x)^3 \times (4x^2)^2}{2x^4}$

(c) $\frac{e^n \times e^m}{e^p}$

(d) $\frac{6b(a^2b)^2}{t(2k)^2} \div \frac{3(ab)^2}{(2kt^2)^3}$

2. Solution Steps

(a) $\frac{a^4 \times a^3}{a^2}$

We use the rule $a^m \times a^n = a^{m+n}$. Thus, $a^4 \times a^3 = a^{4+3} = a^7$.

Then the expression becomes $\frac{a^7}{a^2}$.

We use the rule $\frac{a^m}{a^n} = a^{m-n}$. Thus, $\frac{a^7}{a^2} = a^{7-2} = a^5$.

(b) $\frac{(3x)^3 \times (4x^2)^2}{2x^4}$

We use the rule $(ab)^n = a^n b^n$.

$(3x)^3 = 3^3 x^3 = 27x^3$.

$(4x^2)^2 = 4^2 (x^2)^2 = 16x^4$.

Thus, the expression becomes $\frac{27x^3 \times 16x^4}{2x^4}$.

$27 \times 16 = 432$, so we have $\frac{432x^3 \times x^4}{2x^4}$.

Using the rule $a^m \times a^n = a^{m+n}$, we get $\frac{432x^7}{2x^4}$.

Then $\frac{432}{2} = 216$, and using the rule $\frac{a^m}{a^n} = a^{m-n}$, we get $\frac{x^7}{x^4} = x^{7-4} = x^3$.

Therefore, the expression simplifies to $216x^3$.

(c) $\frac{e^n \times e^m}{e^p}$

We use the rule $a^m \times a^n = a^{m+n}$. Thus, $e^n \times e^m = e^{n+m}$.

The expression becomes $\frac{e^{n+m}}{e^p}$.

We use the rule $\frac{a^m}{a^n} = a^{m-n}$. Thus, $\frac{e^{n+m}}{e^p} = e^{n+m-p}$.

(d) $\frac{6b(a^2b)^2}{t(2k)^2} \div \frac{3(ab)^2}{(2kt^2)^3}$

We can rewrite the division as multiplication by the reciprocal:

$\frac{6b(a^2b)^2}{t(2k)^2} \times \frac{(2kt^2)^3}{3(ab)^2}$

Simplify each term:

$(a^2b)^2 = (a^2)^2 b^2 = a^4 b^2$

$(2k)^2 = 4k^2$

$(2kt^2)^3 = 2^3 k^3 (t^2)^3 = 8k^3 t^6$

$(ab)^2 = a^2b^2$

Substitute these back into the expression:

$\frac{6b(a^4b^2)}{t(4k^2)} \times \frac{8k^3 t^6}{3(a^2b^2)} = \frac{6ba^4b^2}{4tk^2} \times \frac{8k^3 t^6}{3a^2b^2} = \frac{6a^4b^3}{4tk^2} \times \frac{8k^3 t^6}{3a^2b^2}$

Multiply the fractions:

$\frac{6 \times 8 \times a^4 \times b^3 \times k^3 \times t^6}{4 \times 3 \times t \times k^2 \times a^2 \times b^2} = \frac{48a^4b^3k^3t^6}{12a^2b^2k^2t}$

Simplify the expression:

$\frac{48}{12} = 4$

$\frac{a^4}{a^2} = a^2$

$\frac{b^3}{b^2} = b$

$\frac{k^3}{k^2} = k$

$\frac{t^6}{t} = t^5$

Thus, the expression simplifies to $4a^2bkt^5$.

3. Final Answer

(a) $a^5$

(b) $216x^3$

(c) $e^{n+m-p}$

(d) $4a^2bkt^5$