simplify:\ 8\sqrt{44}+7\sqrt{28}+9\sqrt{99}+3\sqrt{112}

To simplify the expression \(8\sqrt{44} + 7\sqrt{28} + 9\sqrt{99} + 3\sqrt{112}\), we'll start by simplifying each square root term.

### Step 1: Simplify each square root.

1. \(\sqrt{44} = \sqrt{4 \times 11} = \sqrt{4} \times \sqrt{11} = 2\sqrt{11}\)
2. \(\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} = 2\sqrt{7}\)
3. \(\sqrt{99} = \sqrt{9 \times 11} = \sqrt{9} \times \sqrt{11} = 3\sqrt{11}\)
4. \(\sqrt{112} = \sqrt{16 \times 7} = \sqrt{16} \times \sqrt{7} = 4\sqrt{7}\)

### Step 2: Substitute back into the expression.

\[
8\sqrt{44} = 8 \times 2\sqrt{11} = 16\sqrt{11}
\]
\[
7\sqrt{28} = 7 \times 2\sqrt{7} = 14\sqrt{7}
\]
\[
9\sqrt{99} = 9 \times 3\sqrt{11} = 27\sqrt{11}
\]
\[
3\sqrt{112} = 3 \times 4\sqrt{7} = 12\sqrt{7}
\]

### Step 3: Combine like terms.

The expression becomes:
\[
16\sqrt{11} + 27\sqrt{11} + 14\sqrt{7} + 12\sqrt{7}
\]

Combine the \(\sqrt{11}\) and \(\sqrt{7}\) terms:
\[
(16 + 27)\sqrt{11} + (14 + 12)\sqrt{7}
\]
\[
43\sqrt{11} + 26\sqrt{7}
\]

### Final Answer:

The simplified expression is \(43\sqrt{11} + 26\sqrt{7}\).

Would you like more details on any part of the process? Here are some related questions:

1. How do you determine the largest perfect square factor when simplifying radicals?
2. Why is it important to combine like terms in expressions with square roots?
3. What is the difference between simplifying and rationalizing square roots?
4. How do you know which terms can be combined in an expression with radicals?
5. Can you simplify the result any further, and under what circumstances?

**Tip:** Always group and combine like terms when simplifying expressions with square roots to achieve the most reduced form.

Learn how to simplify the expression 8sqrt(44) + 7sqrt(28) + 9sqrt(99) + 3sqrt(112) step-by-step. This problem involves simplifying square roots using basic rules and combining like terms. Suitable for Grades 9-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation