Question

Rational exponents practice
Rational, and irrational numbers

Answer 1

Sure! Let's explore rational exponents, as well as rational and irrational numbers through a few practice problems and explanations.

### Rational Exponents

A rational exponent is an exponent that is a fraction. The general form is:

\[
a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m
\]

Here, \(a\) is a real number, \(m\) is an integer, and \(n\) is a positive integer.

### Examples

1. **Convert the Expression:**
Convert \(8^{\frac{2}{3}}\) to a radical form and simplify.

**Solution:**
\[
8^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4
\]

2. **Evaluate the Expression:**
Evaluate \(27^{\frac{1}{3}}\).

**Solution:**
\[
27^{\frac{1}{3}} = \sqrt[3]{27} = 3
\]

3. **Simplification with Rational Exponents:**
Simplify \(x^{\frac{5}{2}} \cdot x^{\frac{3}{2}}\).

**Solution:**
\[
x^{\frac{5}{2}} \cdot x^{\frac{3}{2}} = x^{\frac{5+3}{2}} = x^{\frac{8}{2}} = x^4
\]

4. **Combine and Simplify:**
Simplify \(\frac{16^{\frac{3}{4}}}{4^{\frac{1}{2}}}\).

**Solution:**
\[
16^{\frac{3}{4}} = (4^2)^{\frac{3}{4}} = 4^{2 \cdot \frac{3}{4}} = 4^{\frac{3}{2}} \\
4^{\frac{1}{2}} = \sqrt{4} = 2 \\
\frac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}} = 4^{\frac{3}{2} - \frac{1}{2}} = 4^{\frac{2}{2}} = 4^1 = 4
\]

### Rational and Irrational Numbers

- **Rational Numbers** are numbers that can be expressed as the quotient of two integers (where the denominator is not zero). Examples include \( \frac{1}{2}, 3, -4, 0.75\).

- **Irrational Numbers** are numbers that cannot be expressed as a fraction of two integers. Their decimal form is non-repeating and non-terminating. Examples include \( \pi, \sqrt{2}, e\).

### Identifying Rational and Irrational Numbers

1. Is the number \( \frac{-7}{3} \) rational or irrational?

**Answer:**
Rational (it can be expressed as a fraction)

2. Is \( \sqrt{16} + 1 \) rational or irrational?

**Answer:**
Rational (since \( \sqrt{16} = 4 \) and \( 4 + 1 = 5\))

3. Is \( \sqrt{2} \times \sqrt{4} \) rational or irrational?

**Answer:**
Rational (since \( \sqrt{4} = 2 \) and \( \sqrt{2} \times 2 = 2\sqrt{2} \) is still irrational)

### Practice Problems

Try to solve these on your own:

1. Simplify \( (x^{\frac{2}{3}})^3 \).
2. Evaluate \( 64^{\frac{1}{2}} \).
3. Identify whether \( 0.333... \) (the repeating decimal) is rational or irrational.
4. Convert \( 81^{\frac{4}{4}} \) to its simplest form.
5. Simplify \( \frac{9^{\frac{3}{2}}}{3^{\frac{5}{2}}} \).

Feel free to ask for help or clarification on any of these topics!