1. Simplify
a. 44 x 5-4
\begin{align*} 4^{4} \times 5^{-4} \end{align*}
Solution:
\begin{align*} \text{Formula } \\ & \boxed{x^m \times y^m = (x \times y)^m} \\ \\ &= 4^4 \times 5^{-4} \\ \\ &= 4^4 \times \left(\frac{1}{5}\right)^{4} \\ \\ &= \left(4 \times \frac{1}{5}\right)^{4} \\ \\ &= \left(\frac{4}{5}\right)^{4} \\ \\ &= \frac{256}{625} \\ \\ \end{align*}
b. 22 x (-1/3)2
\begin{align*} 2^2 \times \left(-\frac{1}{3}\right)^2 \end{align*}
Solution:
\begin{align*} \text{Formula: } \\ & \boxed{x^m \times y^m = (x \times y)^m} \\ \\ &= 2^2 \times \left(-\frac{1}{3}\right)^2 \\ \\ &= 2^2 \times \left(-\frac{1}{3}\right)^2 \\ \\ &= \left(2 \times \left(-\frac{1}{3}\right)\right)^2 \\ \\ &= \left(-\frac{2}{3}\right)^2 \\ \\ &= \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right) \\ \\ &= \frac{4}{9} \\ \\ \end{align*}
c. (-2/3)3 x (-3/5)3
\begin{align*} \left(-\frac{2}{3}\right)^3 \times \left(-\frac{3}{5}\right)^3 \\ \\ \end{align*}
Solution:
\begin{align*} \text{Formula: } \\ & \boxed{x^m \times y^m = (x \times y)^m} \\ \\ &= \left(-\frac{2}{3}\right)^3 \times \left(-\frac{3}{5}\right)^3 \\ \\ &= \left(\left(-\frac{2}{3}\right) \times \left(-\frac{3}{5}\right)\right)^3 \\ \\ &= \left(\frac{2}{3} \times \frac{3}{5}\right)^3 \\ \\ &= \left(\frac{2 \times \cancel3}{\cancel3 \times 5}\right)^3 \\ \\ &= \left(\frac{2}{5}\right)^3 \\ \\ &= \frac{8}{125} \\ \\ \end{align*}
d. (1/2)-2 x (2/5)-2
\begin{align*} \left(\frac{1}{2}\right)^{-2} \times \left(\frac{2}{5}\right)^{-2} \end{align*}
Solution:
\begin{align*} \text{Formula: } \\ & \boxed{x^m \times y^m = (x \times y)^m} \\ \\ &= \left(\frac{2}{1}\right)^2 \times \left(\frac{5}{2}\right)^2 \\ \\ &= \left(\cancel2 \times \frac{5}{\cancel2}\right)^2 \\ \\ &= \left(5\right)^2 \\ \\ &= 25 \\ \\ \end{align*}
e. (-5/6)4 ÷ (-7/6)4
\begin{align*} \left(-\frac{5}{6}\right)^4 \div \left(-\frac{7}{6}\right)^4 \\ \\ \end{align*}
Solution:
\begin{align*} &= \left(-\frac{5}{6}\right)^4 \div \left(-\frac{7}{6}\right)^4 \\ \\ &= \left(\frac{5}{\cancel6} \times \frac{\cancel6}{7}\right)^4 \\ \\ &= \left(\frac{5}{7}\right)^4 \\ \\ &= \left(\frac{5^4}{7^4}\right) \\ \\ &= \frac{625}{2401} \\ \\ \end{align*}
f. (-2/3)-5 x (-3/2)-5
\begin{align*} \left(-\frac{2}{3}\right)^{-5} \times \left(-\frac{3}{2}\right)^{-5} \\ \\ \end{align*}
Solution:
\begin{align*} &= \left(-\frac{3}{2}\right)^{5} \times \left(-\frac{2}{3}\right)^{5} \\ \\ &= \left(\left(-\frac{3}{2}\right) \times \left(-\frac{2}{3}\right)\right)^{5} \\ \\ &= \left(\frac{\cancel3}{\cancel2} \times \frac{\cancel2}{\cancel3}\right)^{5} \\ \\ &= (1)^{5} \\ \\ &= 1 \\ \\ \end{align*}
2. Find the value of x so that-
a. 32 x (-4)2 = (-12)2x
\begin{align*} & 3^2 \times (-4)^2 = (-12)^{2x} \\ \end{align*}
Solution:
\begin{align*} \text{Formula: } \\ & \boxed{x^m \times y^m = (x \times y)^m} \\ \\ 3^2 \times (-4)^2 &= (-12)^{2x} \\ \\ (3 \times -4)^2 &= (-12)^{2x} \\ \\ (-12)^2 &= (-12)^{2x} \\ \\ \text{Since bases are same, } & \text{their exponents must be equal.} \\ \\ 2x &= 2 \\ \\ x &= 1 \\ \end{align*}
b. (-3/2)6 x (4/9)3 = (1/2)3x
\begin{align*} \left(-\frac{3}{2}\right)^6 \times \left(\frac{4}{9}\right)^3 = \left(\frac{1}{2}\right)^{3x} \end{align*}
Solution:
\begin{align*} \left(-\frac{3}{2}\right)^6 \times \left(\frac{4}{9}\right)^3 &= \left(\frac{1}{2}\right)^{3x} \\ \left(-\frac{3}{2}\right)^6 \times \left(\frac{2^2}{3^2}\right)^3 &= \left(\frac{1}{2}\right)^{3x} \\ \left(-\frac{3}{2}\right)^6 \times \left[\left(\frac{2}{3}\right)^2\right]^3 &= \left(\frac{1}{2}\right)^{3x} \\ \left(-\frac{3}{2}\right)^6 \times \left(\frac{2}{3}\right)^6 &= \left(\frac{1}{2}\right)^{3x} \\ \left(-\frac{\cancel3}{\cancel2} \times \frac{\cancel2}{\cancel3}\right)^6 &= \left(\frac{1}{2}\right)^{3x} \\ (-1)^6 &= \left(\frac{1}{2}\right)^{3x} \\ 1 & = \left(\frac{1}{2}\right)^{3x} \\ \left(\frac{1}{2}\right)^{0} &= \left(\frac{1}{2}\right)^{3x} \\ \text{Since bases are same, } & \text{their exponents must be equal.} \\ 3x &= 0 \\ x &= 0 \\ \end{align*}
c. (4/5)-2 ÷ (-4/5)-2 = (1)3x
\begin{align*} \left(\frac{4}{5}\right)^{-2} \div \left(-\frac{4}{5}\right)^{-2} &= 1^{3x} \end{align*}
Solution:
\begin{align*} \left(\frac{5}{4}\right)^{2} \div \left(-\frac{5}{4}\right)^{2} &= 1^{3x} \\ \left[\frac{5}{4} \div \left(-\frac{5}{4}\right)\right]^{2} &= 1^{3x} \\ \left[\frac{\cancel5}{\cancel4} \times \left(-\frac{\cancel4}{\cancel5}\right)\right]^{2} &= 1^{3x} \\ (-1)^{2} &= 1^{3x} \\ \\ (-1) \times (-1) &= 1^{3x} \\ \\ 1^1 &= 1^{3x} \\ \\ \text{Since bases are same, } & \text{their exponents must be equal.} \\ \\ 3x & = 1 \\ \\ x & = \frac{1}{3} \\ \\ \end{align*}
d. (9/4)3 x (8/9)3 = 26x
\begin{align*} \left(\frac{9}{4}\right)^3 \times \left(\frac{8}{9}\right)^3 &= 2^{6x} \\ \end{align*}
Solution:
\begin{align*} \left(\frac{\cancel9}{\cancel4} \times \frac{\cancelto{2}8}{\cancel9}\right)^3 &= 2^{6x} \\ \\ 2^3 &= 2^{6x} \\ \\ \text{Since bases are same, } & \text{their exponents must be equal.} \\ \\ 6x &= 3 \\ \\ x &= \frac{3}{6} \\ \\ x &= \frac{1}{2} \\ \\ \end{align*}
e. (15/4)3 ÷ (5/4)3 = 3x
\begin{align*} \left(\frac{15}{4}\right)^3 \div \left(\frac{5}{4}\right)^3 &= 3^x \\ \end{align*}
Solution:
\begin{align*} \left(\frac{15}{4} \div \frac{5}{4}\right)^3 &= 3^x \\ \\ \left(\frac{\cancelto{3}{15}}{\cancel4} \times \frac{\cancel4}{\cancel5}\right)^3 &= 3^x \\ \\ 3^3 &= 3^x \\ \\ \text{Since bases are same, } & \text{their exponents must be equal.} \\ \\ 3^x &= 3^3 \\ \\ x &= 3 \end{align*}