Rules for Operations with Exponents - jessamine.k12.ky.us

[Pages:3]COLLEGE PREP SECTION 5A ? GETTING READY FOR CHAPTER 5

Objectives: ? Simplify exponential equations using the product rule, the quotient rule, the power rule, and the Law of Exponents. ? Evaluate Exponential expressions with a Zero or negative exponent. ? Convert between Scientific Notation and Decimal Notation. ? Use Scientific Notation to multiply and divide.

NOTATION: in the expression , is called the base, and is called the exponent or power.

Rules for Operations with Exponents

Operation

Multiplying ? add exponents

Dividing ? subtract exponents

Power to a power ? multiply exponents

Power of a product ? exponent applies to each factor (like distributing)

Power of a quotient ? exponent applies to numerator and denominator (like distributing)

Power of a negative quotient ? exponent applies to numerator and denominator (like distributing) This will cause everything inside to switch places.

Negative exponents ? moving the exponential factor to the denominator creates a positive exponent

Zero Exponents ? any number or variable that has a zero exponent is always equal to 1

Formula

?

or

1

Example

?

2 16

5 125

5 5 5

3

3

1 4 4

Note: These power rules assume that the variable does not equal 0 whenever it's in the denominator or if it is raised to the zero power.

BEWARE of these common mistakes!!!

Error Formula Description Exponents do not distribute over addition or subtraction!

Example 3 5 3 5

0

Like terms in fractions do not cancel! (Only factors cancel.)

Zero exponent does not mean the same as multiply by zero! Negative exponents do not make a monomial negative!

25 2 353

5 0 4 4

EXAMPLES: Simplify each expression

Product Rule:

Remember to deal with the coefficients separately.

A) ? 3 3 243

B) ? 2 ? 5 ? 10

Quotient Rule:

Coefficients (numbers) are divided, exponents are subtracted.

C)

6 6 216

D)

?

Zero Exponent Rule: Anything with an exponent of zero should be changed to a 1

E) 1

F) 181 18

Negative Exponent Rule: Move ONLY the variable that the exponent is attached to. If it's outside

parentheses, move everything within the parentheses.

G)

H)

I)

J)

? ? ?

Power Rule: If you raise a power to a power, you are multiplying it by itself, therefore, you must raise any

coefficient to the power outside the parentheses, and multiply all exponents.

K)

3 ? 3 ? 3 ? 3 3? 3 L)

7? 7 1

M) 2 16

N)

4 ? ?

O)

P)

?

?

SCIENTIFIC NOTATION. A number is written in scientific notation when it is in the form

10

where 1 || 10 and is an integer.

To change a decimal to scientific notation:

Step 1:

Count the number N of decimal places that the decimal point must be moved in order to get

only one digit () in front of the decimal.

Step 2:

If you had to move the decimal to the left (you started with a large number), then your

exponent is positive ( 10). If you had to move the decimal to the right (you started with a decimal), then your exponent will be negative ( 10.

Examples: Write the following in scientific notation. Q) , 2.384 10

Move the decimal 5 places to the left (the original number is greater than 1), so the exponent is +5.

R)

. 7.1 10

Move the decimal 2 places to the right (the initial number is less than 1), so the exponent is -2.

REVERSING THE PROCESS (going from scientific notation to decimal notation): Look at the exponent on the 10. If the exponent is negative, move the decimal N spaces to the left (toward

the negative end of the number line). If the exponent is positive, move the decimal N spaces to the right (toward the positive end of the number line).

Examples: Write the following in decimal notation.

S)

. 28,000

The positive exponent means the decimal moves to the

right.

T)

. 0.0000149

The negative exponent moves the decimal to the left.

MULTIPLYING & DIVIDING WITH SCIENTIFIC NOTATION. Follow the usual rules of exponents, except separate the pieces. Simplify the numbers, then add/subtract the

exponents on the 10's.

Examples: U) 3 ? 2 10 ? 10 6 10

V)

. . 3.2 ? 4.8 10 ? 10 15.36 10 1.536 10

W)

. .

. 10 2 10

.

X)

. .

.. 10

10

.5 10

5 10

Homework: page 351: # 19-24, 31, 33, 37, 41, 45, 51, 57, 65, 69, 71, 81, 83, 85, 87, 91, 101, 105, 115, 117, 121, 131.

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