https://file.aiquickdraw.com/mathbot/file/1727627401811iwtcotb8.jpg

Let's go through each of the problems step by step.

### a. \( \sqrt{2} \cdot \sqrt{3} \)
Using the property of square roots:
\[
\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}
\]
\[
\sqrt{2} \cdot \sqrt{3} = \sqrt{2 \cdot 3} = \sqrt{6}
\]

### b. \( \sqrt{5} \cdot \sqrt{5} \)
This is simplified using:
\[
\sqrt{a} \cdot \sqrt{a} = a
\]
\[
\sqrt{5} \cdot \sqrt{5} = 5
\]

### c. \( \sqrt{6} \cdot \sqrt{5} \)
Applying the same square root multiplication rule:
\[
\sqrt{6} \cdot \sqrt{5} = \sqrt{6 \cdot 5} = \sqrt{30}
\]

### d. \( 2 \sqrt{2} \cdot 4 \sqrt{3} \)
First, multiply the numerical coefficients \(2 \cdot 4 = 8\), then the square roots:
\[
\sqrt{2} \cdot \sqrt{3} = \sqrt{2 \cdot 3} = \sqrt{6}
\]
So, the full expression becomes:
\[
2 \sqrt{2} \cdot 4 \sqrt{3} = 8 \sqrt{6}
\]

### e. \( 5^3 \cdot \sqrt{6} \cdot 6^3 \cdot \sqrt{5} \)
First, simplify the powers and products:
\[
5^3 = 125, \quad 6^3 = 216
\]
Now multiply the square roots:
\[
\sqrt{6} \cdot \sqrt{5} = \sqrt{6 \cdot 5} = \sqrt{30}
\]
So, the full expression becomes:
\[
125 \cdot 216 \cdot \sqrt{30}
\]
We can calculate:
\[
125 \cdot 216 = 27000
\]
Thus, the result is:
\[
27000 \sqrt{30}
\]

### f. \( \frac{\sqrt[3]{10}}{\sqrt[3]{5}} \)
This can be simplified as:
\[
\frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}}
\]
\[
\frac{\sqrt[3]{10}}{\sqrt[3]{5}} = \sqrt[3]{\frac{10}{5}} = \sqrt[3]{2}
\]

### g. \( \frac{\sqrt{24}}{\sqrt{6}} \)
Using the property of square roots:
\[
\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}
\]
\[
\frac{\sqrt{24}}{\sqrt{6}} = \sqrt{\frac{24}{6}} = \sqrt{4} = 2
\]

### h. \( \frac{10 \sqrt{6}}{5 \sqrt{2}} \)
First, simplify the numerical coefficients:
\[
\frac{10}{5} = 2
\]
Now, simplify the square roots:
\[
\frac{\sqrt{6}}{\sqrt{2}} = \sqrt{\frac{6}{2}} = \sqrt{3}
\]
So, the full expression becomes:
\[
2 \cdot \sqrt{3} = 2\sqrt{3}
\]

---

Let me know if you need any additional details!

Here are 5 related questions:
1. How do you simplify expressions involving cube roots like in problem (f)?
2. Can you apply the same square root rules for fractional powers, such as \( \sqrt{x^2} \)?
3. What is the general rule for multiplying or dividing radical expressions?
4. How do you deal with exponents in radical expressions, such as \( 5^3 \)?
5. What are some strategies for simplifying nested radicals like \( \sqrt{a} \cdot \sqrt{b} \)?

**Tip:** Always look for opportunities to simplify both numerical coefficients and radicals separately to make problems easier to solve!

Solve the following mathematical problems:
a. \( \sqrt{2} \cdot \sqrt{3} \)
b. \( \sqrt{5} \cdot \sqrt{5} \)
c. \( \sqrt{6} \cdot \sqrt{5} \)
d. \( 2 \sqrt{2} \cdot 4 \sqrt{3} \)
e. \( 5^3 \cdot \sqrt{6} \cdot 6^3 \cdot \sqrt{5} \)
f. \( \frac{\sqrt[3]{10}}{\sqrt[3]{5}} \)
g. \( \frac{\sqrt{24}}{\sqrt{6}} \)
h. \( \frac{10 \sqrt{6}}{5 \sqrt{2}} \)

Learn how to simplify radical expressions, including multiplying square roots, cube roots, and dealing with radicals in fractions. This tutorial covers algebraic concepts such as simplifying expressions like \( \sqrt{6} \cdot \sqrt{5} \) and \( \frac{\sqrt{24}}{\sqrt{6}} \). Suitable for high school students (Grades 9-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation