Exponents and Roots

Expressions that contain roots and expressions are found frequently in algebra, and also in some advanced math. By mastering the concepts behind these topics, students will be able to work with ease with such expressions. Again the students will have a second opportunity to:

1) Review and learn Exponent Properties and Algebraic Expressions.

2) Multiplying/Dividing with Exponents

3) Roots and Radicals

4) Simplifying The Square Root of A Product

5) Simplifying Square Roots of Powers

6) Simplifying The Square Root Of A Fraction

7) Rationalize the Denominator Of A Fraction

8) Scientific Notation

9) Quiz On Exponents and Roots

An exponent is a small number written in the upper right of a number, which is referred as the base. Therefore, the exponent tells us how many times that the base will be multiplied by itself. For example, let's consider this multiplication problem:

4 × 4 × 4 × 4 × 4 × 4

In this example, we noticed that the same number is multiplied 5 times, and instead of multiplying the number 4, which is the base of the exponent, five times by itself.

It will be simpler,faster to use the exponential form for: 4 x 4 x 4 x 4 x 4, we just write it this way 45 and it will mean the same thing.

When reading 45, we say four to the fifth power, or 4 raised to the fifth power.

Let’s evaluate the following expressions:

35 = 3 × 3 × 3 x 3 x 3

64 = 6 × 6 × 6 × 6

62 = 6 × 6

In all, exponents have many useful properties. They are:

Property 1

Case 1

am . an = am+n

When multiplying two numbers whose bases are the same, we add their exponents.

This property is also known as Product Rule

Examples

43. 42 = 45

54. 53 = 57

Property 2

Case 2

am/an = am-n

When dividing two numbers whose bases are the same, we subtract their exponents. This property is also known as the Quotient Rule

Examples

45/43 = 45-3 = 42

y6/y4 = y6-4 = y2

Property 3

Case 3

(am)n = amn

If a number is raised to a power which is itself is raised to another power, we multiply the exponents. This property is known as Power Rule I

Examples

(a3)4 = a4.3 = a12

(y2)5 = y2.5 = y10

Property 4

Case 4

(ab)n = anbn

This property is known as Product to power to distribute to each base. By working with this property, we can take a product followed by the power or take the powers followed by the product.

Examples

Use Property 4 to rewrite the following expressions

(4x3y)2 = 42(x)3y2 = 16x6y2

(x2y)4 =(x2)4y4 = x8y4

Property of Zero

Case 5

a0 = 1

Any nonzero number raised to zero power is equal to one

Example

9/9 =1, however 9 = 32. Therefore, you have 32 / 32. Remember Property 2, which stated that when dividing two numbers whose bases are the same, we subtract their exponents. So we have 32 -2 = 30

Thus, 9/9 = 1 = 90

Property 6

Case 6

a-1 = 1/a

This property states that a-1 means the same as the reciprocal of a.

Examples

4-1 = 1/4

x-1 = 1/x

Property 7

Case 7

1/a-m=am

This property states that to solve this expression,just flip and change the sign to positive.

Examples

1/x-3 = x3

1/x-5 = x-5