5.1 Use Properties of Exponents

Scientific notation

A number is expressed in scientific notation if it is in the form c´ 10ⁿ where 1 £ c< 10 and n is an integer.

PROPERTIES OF EXPONENTS

Let a and b be real numbers and let m and n be integers.

Product of Powers Property a^m · aⁿ = a^{__m + n__}

Power of a Power Property (a^m)ⁿ = a^__mn__

Power of a Product Property (ab)^m =a ^_m_ b^_m_

Negative Exponent Property

Zero Exponent Property a⁰ = _1_ ,a ¹ 0

Quotient of Powers Property

Power of a Quotient Property

Example 1

Evaluate numerical expressions

a. (6²)³ = 6 ^{2 · 3} = 6 ⁶ = _46,656_

b.

c.

Example 2

Use scientific notation in real life

Iceland Iceland covers about 1.03 ´ 10⁵ square kilometers and has approximately 2.94 ´ 10⁵ people. About how many people are there per square kilometer?

Solution

Divide population by land area.

Quotient of powers property

Use a calculator.

= _2.85_ Zero exponent property

There are about _3_ people per square kilometer.

Example 3

Simplify expressions

	Power of a product property
	Power of a power property
=_x^15^-^15y^6^-^8_	Quotient of powers property
=_x^0y^-^2_	Simplify exponents.
=_y ^-^2_	Zero exponent property
	Negative exponent property
	Power of a quotient property

	Power of a power property
	Negative exponent property

Example 4

Compare real-life volumes

Beach Ball The radius of a beach ball is about 5.6 times greater than the radius of a baseball. How many times as great as the baseball's volume is the beach ball's volume?

Solution

Let r represent the radius of the baseball.

	The volume of a sphere is.
	Power of a product property
=__5.6^3r^0_	Quotient of powers
=_5.6^3_	Zero exponent property
»176_	Approximate power.

The beach ball's volume is about 176 times as great as the baseball's volume.