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In Exercises 47-52, simplify the expression. Assume that the variables in the denominators are nonzero.

47. [tex]\frac{x^4 y^3}{x^2 y^5}[/tex]

48. [tex]\frac{\left(3 x^2\right)^2 y^4}{3 y^2}[/tex]


Sagot :

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]