Sure! Let's simplify each expression step-by-step:
### 47. Simplify [tex]\(\frac{x^4 y^3}{x^2 y^5}\)[/tex]
1. Identify the exponents:
- The numerator is [tex]\(x^4 y^3\)[/tex].
- The denominator is [tex]\(x^2 y^5\)[/tex].
2. Simplify the [tex]\(x\)[/tex] terms:
- Use the property [tex]\(\frac{x^a}{x^b} = x^{a-b}\)[/tex].
- So, [tex]\(\frac{x^4}{x^2} = x^{4-2} = x^2\)[/tex].
3. Simplify the [tex]\(y\)[/tex] terms:
- Use the property [tex]\(\frac{y^a}{y^b} = y^{a-b}\)[/tex].
- So, [tex]\(\frac{y^3}{y^5} = y^{3-5} = y^{-2}\)[/tex].
4. Combine the simplified terms:
- The simplified form of [tex]\(\frac{x^4 y^3}{x^2 y^5}\)[/tex] is [tex]\(x^2 y^{-2}\)[/tex].
So, the simplified expression is:
[tex]\[
\frac{x^4 y^3}{x^2 y^5} = x^2 y^{-2}
\][/tex]
### 48. Simplify [tex]\(\frac{\left(3 x^2\right)^2 y^4}{3 y^2}\)[/tex]
1. Simplify the numerator:
- The expression in the numerator is [tex]\((3 x^2)^2 y^4\)[/tex].
- First, simplify [tex]\((3 x^2)^2\)[/tex]:
- [tex]\((3 x^2)^2 = (3)^2 (x^2)^2 = 9 x^{2 \cdot 2} = 9 x^4\)[/tex].
- So, the numerator becomes [tex]\(9 x^4 y^4\)[/tex].
2. Simplify the denominator:
- The denominator is [tex]\(3 y^2\)[/tex].
3. Divide the constants:
- The constant [tex]\(9\)[/tex] in the numerator and [tex]\(3\)[/tex] in the denominator:
- [tex]\(\frac{9}{3} = 3\)[/tex].
4. Simplify the [tex]\(x\)[/tex] terms:
- The numerator has [tex]\(x^4\)[/tex] and the denominator has no [tex]\(x\)[/tex] terms.
- So, [tex]\(x^4\)[/tex] remains as is.
5. Simplify the [tex]\(y\)[/tex] terms:
- Use the property [tex]\(\frac{y^a}{y^b} = y^{a-b}\)[/tex].
- So, [tex]\(\frac{y^4}{y^2} = y^{4-2} = y^2\)[/tex].
6. Combine the simplified terms:
- The simplified form is [tex]\(3 x^4 y^2\)[/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]
Therefore, the simplified forms of the given expressions are:
1. [tex]\(\frac{x^4 y^3}{x^2 y^5} = x^2 y^{-2}\)[/tex]
2. [tex]\(\frac{\left(3 x^2\right)^2 y^4}{3 y^2} = 3 x^4 y^2\)[/tex]
