Sure, let's go through each problem step-by-step to simplify the expressions:
### 47. [tex]\(\frac{x^4 y^3}{x^2 y^5}\)[/tex]
This can be simplified by subtracting the exponents of like bases in the numerator and the denominator:
[tex]\[
\frac{x^4 y^3}{x^2 y^5} = x^{4-2} \cdot y^{3-5}
\][/tex]
This simplifies to:
[tex]\[
x^2 \cdot y^{-2}
\][/tex]
Since [tex]\( y^{-2} = \frac{1}{y^2} \)[/tex], the final simplified form is:
[tex]\[
\frac{x^2}{y^2}
\][/tex]
### 48. [tex]\(\frac{(3x^2)^2 y^4}{3y^2}\)[/tex]
First, simplify the expression inside the numerator:
[tex]\[
(3x^2)^2 = 3^2 \cdot (x^2)^2 = 9x^4
\][/tex]
Now substitute this back into the expression:
[tex]\[
\frac{9x^4 y^4}{3 y^2}
\][/tex]
Next, divide the coefficients and simplify the variables using exponent rules:
[tex]\[
\frac{9}{3} \cdot \frac{x^4 y^4}{y^2} = 3 \cdot x^4 \cdot y^{4-2} = 3x^4 y^2
\][/tex]
So, the final simplified form is:
[tex]\[
3x^4 y^2
\][/tex]
### 49. [tex]\(\left(\frac{4}{x^2}\right)^2\)[/tex]
Apply the power to both the numerator and the denominator:
[tex]\[
\left(\frac{4}{x^2}\right)^2 = \frac{4^2}{(x^2)^2} = \frac{16}{x^4}
\][/tex]
The final simplified form is:
[tex]\[
\frac{16}{x^4}
\][/tex]
### 50. [tex]\(\left(\frac{2}{xy}\right)^{-3}\)[/tex]
First, recall that raising a fraction to a negative exponent is equivalent to taking the reciprocal of the fraction and then raising it to the positive exponent:
[tex]\[
\left(\frac{2}{xy}\right)^{-3} = \left(\frac{xy}{2}\right)^3
\][/tex]
Now apply the exponent to both the numerator and the denominator:
[tex]\[
\left(\frac{xy}{2}\right)^3 = \frac{(xy)^3}{2^3} = \frac{x^3 y^3}{8}
\][/tex]
So, the final simplified form is:
[tex]\[
\frac{x^3 y^3}{8}
\][/tex]
In summary:
1. [tex]\(\frac{x^2}{y^2}\)[/tex]
2. [tex]\(3x^4 y^2\)[/tex]
3. [tex]\(\frac{16}{x^4}\)[/tex]
4. [tex]\(\frac{x^3 y^3}{8}\)[/tex]
