Rational Exponents

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×aⁿ=a^m+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. 

(a^m)ⁿ=a^mn

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. 

a^m×b^m=(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 √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.