https://file.aiquickdraw.com/mathbot/file/17267862194587k4724ir.jpg

The problem in the image simplifies the given expression by applying several steps involving radicals and factoring. Let's break down the process as shown:

### Given Expression:
\[
\sqrt{a^5 b^9} - \sqrt{a^7 b^{11}} + \sqrt{a^{13} b}
\]

### Step 1: Factor each radicand using perfect squares

The first step is to factor each term under the square roots into products of perfect squares:
\[
\sqrt{a^5 b^9} = \sqrt{a^4 b^8 \cdot ab}
\]
\[
\sqrt{a^7 b^{11}} = \sqrt{a^6 b^{10} \cdot ab}
\]
\[
\sqrt{a^{13} b} = \sqrt{a^{12} \cdot ab}
\]

### Step 2: Write each term as a product of radicals
\[
\sqrt{a^4 b^8 \cdot ab} = \sqrt{a^4 b^8} \cdot \sqrt{ab} = a^2 b^4 \sqrt{ab}
\]
\[
\sqrt{a^6 b^{10} \cdot ab} = \sqrt{a^6 b^{10}} \cdot \sqrt{ab} = a^3 b^5 \sqrt{ab}
\]
\[
\sqrt{a^{12} \cdot ab} = \sqrt{a^{12}} \cdot \sqrt{ab} = a^6 \sqrt{ab}
\]

Now the expression becomes:
\[
a^2 b^4 \sqrt{ab} - a^3 b^5 \sqrt{ab} + a^6 \sqrt{ab}
\]

### Step 3: Factor out the common terms
The term \(\sqrt{ab}\) is common to all terms, as is \(a\), so we factor out the lowest power of \(a\), which is \(a^2\), along with \(\sqrt{ab}\):

\[
a^2 \sqrt{ab} (b^4 - a b^5 + a^4)
\]

### Final Simplified Expression:
\[
a^2 \sqrt{ab} (b^4 - ab^5 + a^4)
\]

This is the fully simplified form of the original expression.

Would you like to explore any further details, or have any additional questions about this? Here are some related questions:

1. How do you factor perfect squares when dealing with radicals?
2. What is the process of simplifying expressions involving square roots?
3. Can this method be applied to cube roots or higher-order roots?
4. How do you know what common terms to factor out in algebraic expressions?
5. How would the final answer change if the variables \(a\) and \(b\) were negative?

**Tip:** Always look for common factors, especially when dealing with radicals, to simplify complex algebraic expressions.

Simplify the expression assuming all variables represent positive real numbers: √(a^5 b^9) - √(a^7 b^11) + √(a^13 b).

Explore how to simplify complex radical expressions involving powers of variables, using factorization and algebraic simplification. This example walks through the process step by step, ideal for high school students learning advanced algebra concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation