## Key Concepts

- Define a rational exponent.
- Solve equations with rational exponents using the product of powers property.
- Solve equations with rational exponents using the power of a power property.
- Solve equations with rational exponents using the power of a product property.
- Solve equations with rational exponents using the quotient of powers property.

## Rational Exponents

### Fractions

A part of a whole is called a fraction.

- All fractions can be placed on the number line.

### Types of Fractions

### Decimal numbers

The numbers whose whole number part and fractional part are separated by a decimal point are called decimal points.

### Factors and multiples

A **factor** is a number or a group of numbers that are multiplied together to make a product.

A **multiple** is the product of a quantity and a whole number.

### Exponents

Repeated multiplication can be represented in more than one way.

You can use an exponent to write the **repeated multiplication **of a number.

A number that can be written using exponents is called a power.

We read as 2 raised to the power of 3.

#### Rational exponents

When a number p is raised to power 1/2, we can write them as √p.

The expressions with exponents that are rational numbers are called **rational exponents** (also called fractional exponents).

#### Laws of exponents

**Law: **When two terms with the same base are multiplied, the powers are added.

a^{m}×a^{n}=a^{m+n}

**Example: **Evaluate 2^{4} × 2^{9}

**Sol:** 2^{4} × 2^{9 }= 2^{(4+9)}

= 2^{13}

= 8192

**Use the product of powers property to solve equations with rational exponents**

#### Law of exponents

**Law: **When raising a power to a new power, multiply the exponents.

(a^{m})^{n}=a^{mn}

**Example: **Evaluate (5^{3})^{2}

**Sol:** (5^{3})^{2 }= 5^{(3×2)}

= 5^{6}

= 15625

**Use the power of a power property to solve equations with rational exponents **

#### Law of exponents

**Law:** When multiplying expressions with the same exponent but different bases, multiply the bases and use the same exponent.

a^{m}×b^{m}=(a×b)^{m}

**Example: **Evaluate 6^{2}×5^{2}

**Sol:** 6^{2}×5^{2 }= (6×5)^{2}

= 30^{2}

= 900

**Use the power of a product property to solve equations with rational exponents**

#### Law of exponents

**Law:** When dividing two powers with the same base, we subtract the exponents.

**Use the quotient of powers property to solve equations with rational exponents**

## Exercise

- Write the radical √14641 using rational exponents.
- What is the value of x in 27
^{(x/2)}= 3^{(x-1)}? - Solve: 3
^{(x/2+1)}= 3^{(-5x/2)} - If the volume of a sphere is V=4/3 πr
^{3}is equal to 392 m^{3}. Find the radius. - Write the radical √b
^{a}using rational exponent.

### Concept Map

### What we have learned

- Repeated multiplication can be represented in more than one way.
- You can use an exponent to write the
**repeated multiplication**of a number.

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