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## 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.

### 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.

am×an=am+n

Example: Evaluate 24 × 29

Sol: 24 × 29 = 2(4+9)

= 213

= 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.

(am)n=amn

Example: Evaluate (53)2

Sol: (53)2 = 5(3×2)

= 56

= 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.

am×bm=(a×b)m

Example: Evaluate 62×52

Sol: 62×52 = (6×5)2

= 302

= 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 πr3 is equal to 392 m3. Find the radius.
• Write the radical √ba using rational exponent.

### 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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