https://file.aiquickdraw.com/mathbot/file/1726708207565f2uuwz89.jpg

Let's solve the questions in the image step by step:

### 1. Simplify the expressions to their simplest exponential form.

#### a. 
\[
10 \times 10^6 \times 10^{-4} \times 10^{-7}
\]
Using the property of exponents: \( a^m \times a^n = a^{m+n} \)

\[
10^{1 + 6 - 4 - 7} = 10^{-4}
\]

#### b. 
\[
10 : 100^2
\]
First, express 100 as \(10^2\), then:
\[
10 : (10^2)^2 = 10 : 10^4 = 10^{1-4} = 10^{-3}
\]

#### c.
\[
5^3 \times \left( \frac{1}{25} \right)^{-1} \times 5^{-2}
\]
First, convert \(\frac{1}{25} = 5^{-2}\), so it becomes:
\[
5^3 \times (5^{-2})^{-1} \times 5^{-2} = 5^3 \times 5^2 \times 5^{-2}
\]
Using properties of exponents:
\[
5^{3+2-2} = 5^3
\]

#### d.
\[
\left( \frac{1}{3} \right)^5 : 9
\]
First, express 9 as \(3^2\):
\[
\left( \frac{1}{3} \right)^5 : 3^2 = 3^{-5} : 3^2 = 3^{-5 - 2} = 3^{-7}
\]

### 2. Rationalize the square roots.

#### a.
\[
\frac{10}{\sqrt{5}}
\]
Multiply by \(\frac{\sqrt{5}}{\sqrt{5}}\) to rationalize:
\[
\frac{10\sqrt{5}}{5} = 2\sqrt{5}
\]

#### b.
\[
\frac{10}{\sqrt{13} + \sqrt{8}}
\]
Multiply numerator and denominator by \(\sqrt{13} - \sqrt{8}\) to use the difference of squares:
\[
\frac{10(\sqrt{13} - \sqrt{8})}{(\sqrt{13})^2 - (\sqrt{8})^2} = \frac{10(\sqrt{13} - \sqrt{8})}{13 - 8} = 2(\sqrt{13} - \sqrt{8})
\]

#### c.
\[
\frac{20}{\sqrt{10}}
\]
Multiply by \(\frac{\sqrt{10}}{\sqrt{10}}\):
\[
\frac{20\sqrt{10}}{10} = 2\sqrt{10}
\]

#### d.
\[
\frac{24}{\sqrt{13} - \sqrt{7}}
\]
Multiply by \(\frac{\sqrt{13} + \sqrt{7}}{\sqrt{13} + \sqrt{7}}\):
\[
\frac{24(\sqrt{13} + \sqrt{7})}{(\sqrt{13})^2 - (\sqrt{7})^2} = \frac{24(\sqrt{13} + \sqrt{7})}{13 - 7} = 4(\sqrt{13} + \sqrt{7})
\]

#### e.
\[
\frac{15}{\sqrt{7} + \sqrt{2}}
\]
Multiply by \(\frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} - \sqrt{2}}\):
\[
\frac{15(\sqrt{7} - \sqrt{2})}{(\sqrt{7})^2 - (\sqrt{2})^2} = \frac{15(\sqrt{7} - \sqrt{2})}{7 - 2} = 3(\sqrt{7} - \sqrt{2})
\]

#### f.
\[
\frac{0}{\sqrt{3}} = 0
\]

---

Would you like a more detailed explanation for any of these solutions? 

Here are 5 related questions for further practice:

1. Simplify \( 2^5 \times 2^{-3} \times 2^4 \).
2. Rationalize \( \frac{12}{\sqrt{6}} \).
3. Simplify \( \left( \frac{1}{2} \right)^4 \times 2^3 \).
4. Simplify \( 5^3 \times \frac{1}{5^2} \).
5. Rationalize \( \frac{7}{\sqrt{7} + 3} \).

**Tip:** When multiplying terms with the same base, simply add their exponents!

Simplify expressions and rationalize square roots as per the provided mathematical problems.

Learn how to simplify expressions involving exponents and rationalize square roots step by step. This math exercise includes using properties of exponents and the conjugate method for rationalizing square roots. Suitable for students in 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