Evaluate and simplify the following expressions: a) 32 x 23 x 41 x 31 x 42, b) 33 x 44 x 5-231 x 5-1 x 41, c) (31)2, d) Expand (23 x⁻¹ (1)/(32))⁻¹ and simplify. Write your answer as a positive power

Group terms with the same base:

$$(3^2 \times 3^1) \times 2^3 \times (4^1 \times 4^2)$$

Apply the rule $a^m \times a^n = a^{m+n}$:

$$3^{2+1} \times 2^3 \times 4^{1+2}$$ $$3^3 \times 2^3 \times 4^3$$

Calculate the values:

$$3^3 = 3 \times 3 \times 3 = 27$$ $$2^3 = 2 \times 2 \times 2 = 8$$ $$4^3 = 4 \times 4 \times 4 = 64$$

Multiply these values:

$$27 \times 8 \times 64$$ $$216 \times 64$$ $$13824$$ $$\boxed{13824}$$

Group terms with the same base and apply the rule $\frac{a^m}{a^n} = a^{m-n}$:

$$\left(\frac{3^3}{3^1}\right) \times \left(\frac{4^4}{4^1}\right) \times \left(\frac{5^{-2}}{5^{-1}}\right)$$ $$3^{3-1} \times 4^{4-1} \times 5^{-2 - (-1)}$$ $$3^2 \times 4^3 \times 5^{-2+1}$$ $$3^2 \times 4^3 \times 5^{-1}$$

Calculate the values:

$$3^2 = 9$$ $$4^3 = 64$$ $$5^{-1} = \frac{1}{5}$$

Multiply these values:

$$9 \times 64 \times \frac{1}{5}$$ $$576 \times \frac{1}{5}$$ $$\frac{576}{5}$$ $$\boxed{\frac{576}{5}}$$

Apply the rule $(a^m)^n = a^{mn}$:

$$(3^1)^2 = 3^{1 \times 2} = 3^2$$

Calculate the value:

$$3^2 = 3 \times 3 = 9$$ $$\boxed{9}$$

Write your answer as a positive power.
First, rewrite $\frac{1}{3^2}$ as $3^{-2}$:

$$(2^3 x^{-1} 3^{-2})^{-1}$$

Apply the outer exponent of $-1$ to each term inside the parentheses using the rule $(abc)^n = a^n b^n c^n$ and $(a^m)^n = a^{mn}$:

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

Rewrite terms with negative exponents as positive exponents using the rule $a^{-n} = \frac{1}{a^n}$:

$$2^{-3} = \frac{1}{2^3}$$

So the expression becomes:

$$\frac{1}{2^3} \times x \times 3^2$$ $$\frac{x \times 3^2}{2^3}$$ $$\boxed{\frac{9x}{8}}$$