Exponents and Scientific Notation

Exponents and Scientific Notation

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Rational Numbers

Rational numbers

• are ratios of two integers

• if you can convert a number into a fraction it is a rational number

• all positive and negative numbers

• decimals that terminate or have repeating patterns

Examples of Rational numbers

All Fractions are Division problems!

Decimals to Fractions

• numbers to the left of the decimal is the whole of a fractions

• Ex: In the decimal 5.31, the 5 is the whole and the .31 would be used to make the faction 31/100

Decimals!!!

Decimals to Percent

• numbers to the left of the decimal is a hint that the percent will be greater than 100

• Ex: In the decimal 5.31, the percent will be 531% once multiplied by 100

Percents to Fractions

• Percents greater than 100 will have a fraction with a whole number

• Ex: In the percent 567%, the fraction will be 5 67/100 once reduced to simplest form

Converting repeating decimals to fractions

Factoring Expressions

• Find the GCF and put it outside the parentheses

• Divide all terms by the GCF and put those dividends in the parentheses

• Distribute the GCF to all terms in parentheses to check your work

Example #1: Factor 4x - 32

The GCF of 4x and -32 is 4, so 4 goes outside of the parentheses.

When we divide 4x by 4, we get x.

When we divide -32 by 4, we get -8.

All together, that gives us our answer: 4(x - 8)

To check, we can distribute the 4 back into the terms in parentheses.

4(x - 8) = 4x - 32 CHECK

Example #2: Factor 18ab + 42a

The GCF of 18ab and 42a is 6a, so 6a goes outside of the parentheses.

When we divide 18ab by 6a, we get 3b.

When we divide 42a by 6a, we get 7.

All together, that gives us our answer: 6a(3b + 7)

To check, we can distribute the 4 back into the terms in parentheses.

6a(3b + 7) = 18ab + 42a CHECK

Distributive Property

Use this property to simplify expressions like this.

a(b + c) = ab + ac

a(b - c) = ab - ac

When you use the distributive property...

• You multiply the number on the outside of the parenthesis by what is inside.

• You need to use the sign in front of the number to determine if the number is a positive or a negative.

​The Distributive Property

​The distributive property allows us to work around parentheses so that we can simplify expressions!

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To use the distributive property, multiply every term inside of parentheses by the term that is outside of the parentheses.

Scientific Notation

Operations with Scientific Notation

by B. Cox

Multiply and Divide Scientific Notation

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​​Multiplying Numbers in Scientific Notation

​When multiplying numbers written in scientific notation, we follow the procedure below,

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1. ​Multiply the first factors

2. ​Multiply the powers of 10 (add the exponents)

3. ​Rewrite the result in scientific notation (if necessary)

​​Multiplying Numbers in Scientific Notation

​​Dividing Numbers in Scientific Notation

​When dividing numbers written in scientific notation, we follow the procedure below,

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1. ​Divide the first factors

2. Divide the powers of 10(subtract the exponents)

3. ​Rewrite the result in scientific notation (if necessary)

How to Add and Subtract

• Determine the number by which to increase the smaller exponent by so it is equal to the larger exponent.

• Increase the smaller exponent by this number and move the decimal point of the number with the smaller exponent to the left the same number of places. (i.e. divide by the appropriate power of 10.)

• Add or subtract the new decimal numbers. Write the answer with the power of 10 that both numbers have.

• If the answer is not in scientific notation (i.e. if the coefficient is not between 1 and 10) convert it to scientific notation.

Solution

• We need to move the decimal to the left on smaller exponent and increase the power by 1.

• When we do that 6.2 x 10⁴ = .62 x 10⁵

• Now add the first numbers and keep the power of 10.

• 3.4 x 10⁵ + .62 x 10⁵ = 4.02 x 10⁵

• That is is scientific notation, so I am done.

• 3.4 x 10⁵ + 6.2 x 10⁴

Example 1 (addition):

Solution

• We need to increase the smaller exponent by 2 by moving the decimal 2 places left.

• When we do that we get 3.9 x 10⁷ = .039 x 10⁹

• Now we subtract the two first numbers and keep the power of 10.

• 5.1 x 10⁹ - .039 x 10⁹ = 5.061 x 10⁹

• That is in scientific notation, so I am done.

• 5.1 x 10⁹ - 3.9 x 10⁷

Example 2 (subtraction):