https://file.aiquickdraw.com/mathbot/file/17259919263823nyr6w1b.jpg

Let's solve each expression step by step. 

### (a) \(\sqrt{200} - \sqrt{32}\)

We can simplify both square roots:

- \(\sqrt{200} = \sqrt{4 \times 50} = 2\sqrt{50} = 2\sqrt{25 \times 2} = 2 \times 5 \times \sqrt{2} = 10\sqrt{2}\)
- \(\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}\)

Now subtract the two terms:

\[
10\sqrt{2} - 4\sqrt{2} = 6\sqrt{2}
\]

**Answer: \(6\sqrt{2}\)**

---

### (b) \((3a^3b^3)(4ab^2)^2\)

First, simplify the powers of the second term:

\[
(4ab^2)^2 = 16a^2b^4
\]

Now multiply the terms:

\[
(3a^3b^3) \times (16a^2b^4) = 48a^{3+2}b^{3+4} = 48a^5b^7
\]

**Answer: \(48a^5b^7\)**

---

### (c) \(\left(\frac{3x^3y^{-3}}{x^{-2}y^{1/2}}\right)^{-2}\)

First, simplify inside the parentheses:

\[
\frac{3x^3y^{-3}}{x^{-2}y^{1/2}} = 3x^{3 - (-2)}y^{-3 - 1/2} = 3x^{5}y^{-7/2}
\]

Now apply the \(-2\) exponent:

\[
\left(3x^5y^{-7/2}\right)^{-2} = 3^{-2}x^{-10}y^{7}
\]

\[
= \frac{1}{9x^{10}y^{-7}} = \frac{y^7}{9x^{10}}
\]

**Answer: \(\frac{y^7}{9x^{10}}\)**

---

### (d) \(\frac{x^2 + 3x + 2}{x^2 - x - 2}\)

Factor both the numerator and denominator:

- \(x^2 + 3x + 2 = (x + 1)(x + 2)\)
- \(x^2 - x - 2 = (x - 2)(x + 1)\)

Now simplify the fraction:

\[
\frac{(x + 1)(x + 2)}{(x - 2)(x + 1)} = \frac{x + 2}{x - 2}
\]

**Answer: \(\frac{x + 2}{x - 2}\)**

---

### (e) \(\frac{\frac{x^2}{x^2 - 4}}{\frac{x + 1}{x + 2}}\)

The denominator \(x^2 - 4\) can be factored as \((x - 2)(x + 2)\). So we rewrite the expression:

\[
\frac{\frac{x^2}{(x - 2)(x + 2)}}{\frac{x + 1}{x + 2}} = \frac{x^2}{(x - 2)(x + 2)} \times \frac{x + 2}{x + 1}
\]

Cancel out the \((x + 2)\):

\[
\frac{x^2}{(x - 2)} \times \frac{1}{x + 1} = \frac{x^2}{(x - 2)(x + 1)}
\]

**Answer: \(\frac{x^2}{(x - 2)(x + 1)}\)**

---

### (f) \(\frac{\frac{y}{x} - \frac{x}{y}}{\frac{1}{x} + \frac{1}{y}}\)

First, simplify the numerator:

\[
\frac{y}{x} - \frac{x}{y} = \frac{y^2 - x^2}{xy}
\]

Now simplify the denominator:

\[
\frac{1}{x} + \frac{1}{y} = \frac{y + x}{xy}
\]

Now divide the two expressions:

\[
\frac{\frac{y^2 - x^2}{xy}}{\frac{y + x}{xy}} = \frac{y^2 - x^2}{y + x} = \frac{(y - x)(y + x)}{y + x}
\]

Cancel out the \((y + x)\):

\[
y - x
\]

**Answer: \(y - x\)**

---

Do you need further clarification or details on any part? Here are some related questions:

1. How do you simplify square roots in general?
2. What are the properties of exponents used in simplification?
3. How can you factor quadratic expressions efficiently?
4. How do you simplify complex rational expressions?
5. How do you handle negative exponents in fractions?

**Tip**: Always check for factoring opportunities in both the numerator and the denominator before simplifying rational expressions!

Simplify each expression. Write your final answer without negative exponents.

This exercise covers the simplification of algebraic expressions involving exponents, radicals, and rational expressions. It includes step-by-step solutions for simplifying expressions such as square roots, factoring quadratic equations, and working with negative exponents. Suitable for high school students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation