Sure, let's go through each exercise step-by-step:
### Exercise 47:
Simplify the expression [tex]\(\frac{x^4 y^3}{x^2 y^5}\)[/tex].
1. Divide the powers of [tex]\(x\)[/tex]:
[tex]\[
\frac{x^4}{x^2} = x^{4-2} = x^2
\][/tex]
2. Divide the powers of [tex]\(y\)[/tex]:
[tex]\[
\frac{y^3}{y^5} = y^{3-5} = y^{-2} = \frac{1}{y^2}
\][/tex]
So, the simplified expression is:
[tex]\[
\frac{x^4 y^3}{x^2 y^5} = \frac{x^2}{y^2}
\][/tex]
### Exercise 48:
Simplify the expression [tex]\(\frac{\left(3 x^2\right)^2 y^4}{3 y^2}\)[/tex].
1. Expand and simplify the numerator [tex]\(\left(3 x^2\right)^2 y^4\)[/tex]:
[tex]\[
(3 x^2)^2 = 3^2 \cdot (x^2)^2 = 9 x^4
\][/tex]
So, the numerator is:
[tex]\[
(9 x^4) y^4 = 9 x^4 y^4
\][/tex]
2. Simplify the fraction [tex]\(\frac{9 x^4 y^4}{3 y^2}\)[/tex]:
[tex]\[
\frac{9 x^4 y^4}{3 y^2} = \frac{9}{3} \cdot \frac{x^4}{1} \cdot \frac{y^4}{y^2} = 3 x^4 y^{4-2} = 3 x^4 y^2
\][/tex]
So, the simplified expression is:
[tex]\[
\frac{\left(3 x^2\right)^2 y^4}{3 y^2} = 3 x^4 y^2
\][/tex]
### Exercise 49:
Simplify the expression [tex]\(\left(\frac{4}{x^2}\right)^2\)[/tex].
1. Square the fraction:
[tex]\[
\left(\frac{4}{x^2}\right)^2 = \frac{4^2}{(x^2)^2} = \frac{16}{x^4}
\][/tex]
So, the simplified expression is:
[tex]\[
\left(\frac{4}{x^2}\right)^2 = \frac{16}{x^4}
\][/tex]
### Exercise 50:
Simplify the expression [tex]\(\left(\frac{2}{x y}\right)^{-3}\)[/tex].
1. Apply the negative exponent:
[tex]\[
\left(\frac{2}{x y}\right)^{-3} = \left(\frac{x y}{2}\right)^3 = \frac{(x y)^3}{2^3} = \frac{x^3 y^3}{8}
\][/tex]
So, the simplified expression is:
[tex]\[
\left(\frac{2}{x y}\right)^{-3} = \frac{x^3 y^3}{8}
\][/tex]
Each of the above steps leads us to the following simplified expressions:
- Exercise 47: [tex]\(\frac{x^2}{y^2}\)[/tex]
- Exercise 48: [tex]\(3 x^4 y^2\)[/tex]
- Exercise 49: [tex]\(\frac{16}{x^4}\)[/tex]
- Exercise 50: [tex]\(\frac{x^3 y^3}{8}\)[/tex]
