Math in Our World. Section 5.6. Exponents and Scientific Notation

Section 5.6

Exponents and

Scientific Notation

Learning Objectives



Define integer exponents.



Use rules for exponents.



Convert between scientific and decimal notation.



Perform operations with numbers in scientific

notation.

Exponents

For any positive integer n,

where a is called the base and n is called the

exponent.

Exponents are a concise method of representing repeated multiplications. For example, 5 x 5 x 5 x 5 can be written as 54_{, the exponent 4 describing how many times to}

multiply 5 by itself. The expression an is read as “a to the nth power.” When the exponent is 2, such as 52_{, it can be}

Exponents

When the exponent is one, it is usually not

written; that is, a

_{= a.}

When an exponent is negative, it is defined as

follows: For any positive integer n,

EXAMPLE 1

Evaluating Expressions with

Exponents

Evaluate each expression

.

(a) 6

_{(b) 3}

_{(c) 9}

SOLUTION

(a) 63 = 6 x 6 x 6 = 216 (b)

Exponents

For any nonzero real numbers a and any integers m and n:

1. The product rule: am _{x a}n _{= a}m + n

When two expressions with the same base are multiplied, you keep the base unchanged and add the exponents.

2. The quotient rule: am = am – n

an

When two expressions with the same base are divided, you keep the base unchanged and subtract the exponents.

3. The power rule: (am₎n _{= a}m x n

When an expression with an exponent is raised to a power, you keep the base unchanged and multiply the two

EXAMPLE 2

Using Rules for Exponents

Simplify the expression using rules for exponents,

and then evaluate the resulting expression.

(a) 3

• 3

(b) 5

EXAMPLE 2

Identify Properties of

Real Numbers

SOLUTION

(a) The bases are the same and we’re multiplying, so we add exponents (product rule).

33 • 35 _{= 3}3 + 5 _{= 3}8 _{which is 6,561}

(b) The bases are the same and we’re dividing, so we subtract the exponents (quotient rule)

(c) When raising an exponential expression to a power, multiply the exponents (power rule).

Scientific Notation

Rules for Writing a Number in Scientific Notation:

Step 1 Move the decimal point either right or left so that there

is exactly one digit to the left of the decimal point.

Step 2 Write the resulting number times 10 to some power:

(a) If the decimal point was moved to the left, the exponent of 10 is the number of places the decimal point was moved. (b) If the decimal point was moved right, the exponent of 10 is the negative of the number of places the decimal point was moved.

EXAMPLE 3

Writing Numbers in

Scientific Notation

Write each number in scientific notation.

(a) 3,572,000,000

(a) Move the decimal point 9 places to the left so that it falls between the 3 and 5.

In scientific notation, 3,572,000,000 = 3.572 x 109_.

(b) Move the point to the right 5 places so that it will fall between the 8 and 7.

In scientific notation, 0.000087 = 8.7 x 10 - 5.

3572000000

EXAMPLE 3

Writing Numbers in

Scientific Notation

SOLUTION

Scientific Notation

Rules for Converting a Number in Scientific Notation to Decimal Notation:

(a) If the exponent of the power of 10 is positive, move the decimal point to the right the same number of places as the exponent. You may need to put in zeros as

placeholders when you run out of digits.

EXAMPLE 4

Converting from Scientific to

Decimal Notation

Write each number in decimal notation.

(a) 4.192 x 10

(a) Since the exponent is positive, move the decimal point 8 places to the right.

In decimal notation, 4.192 x 108 _{= 419,200,000.}

(b) Since the exponent is negative, move the decimal point 8 places to the left.

In decimal notation, 6.37 x 10 – 8 = 0.0000000637

00000000

6.37

0

4.192

0000

EXAMPLE 4

Converting from Scientific to

Decimal Notation

Operations with Numbers in

Scientific Notation

One of the biggest reasons that scientific notation is useful is that it makes multiplication and division of very large and very small numbers very easy. Suppose you are asked to multiply 2 x 107 _{and 4.2 x 10}10_{. In decimal form, this would}

be challenging, to say the least. But let’s write it out and see what happens:

(2 x 107_{) x (4.2 x 10}10_{) = (2 x 4.2) x (10}7 _{x 10}10_{) = 8.4 x 10}17 Regroup

Decimals

Regroup

base 10s The Commutative

Property of Multiplication

EXAMPLE 5

Multiplying Numbers in

Scientific Notation

Find each product. Write your answer in scientific

notation.

(a) Multiply 3 x 2.2 to get 6.6, and add exponents 5 + 3 to get 8.

(3 x 105_{)(2.2 x 10}3_{) = 6.6 x 10}8

(b) This time, when we multiply 5 x 7, we get a two-digit number, so our answer won’t be in proper scientific

notation:

(5 x 102)(7 x 103) = 35 x 105

We can remedy this by moving the decimal point one place left, which raises the exponent by 1:

35 x 105 _{= 3.5 x 10}6

EXAMPLE 5

Multiplying Numbers in

Scientific Notation

EXAMPLE 6

Multiplying Numbers in

Scientific Notation

Find each product. Round the decimal part of your

answer to two decimal places.

(a) (3.25 x 10

)(5.1 x 10

)

(b) (8.6 x 10

_{)(9.7 x 10}

₎

(a) (3.25 x 10 - 4_{)(5.1 x 10} - 3_{) = 16.575 x 10} – 7 ≈ 1.66 x 10 – 6

Be careful when moving the decimal point! Adding 1 to - 7 makes it - 6.

(b) (8.6 x 103_{)(9.7 x 10} - 6_{) = 83.42 x 10} – 3 ≈ 8.34 x 10 - 2

EXAMPLE 7

Dividing Numbers in

Scientific Notation

In each case, we’ll divide the decimal parts, then subtract the exponents to find the appropriate power of 10.

EXAMPLE 7

Dividing Numbers in

Scientific Notation

This time, the answer needs to be rewritten to put it in

proper scientific notation. The decimal point will be moved one place to the right, so we need to subtract one from the exponent.

EXAMPLE 7

Dividing Numbers in

Scientific Notation

EXAMPLE 8

Applying Scientific Notation to

Astronomy

The earth’s orbit around the sun is not circular, but

it’s reasonably close to a circle with radius 9.3

x 10

miles. How far does the earth travel in one year?

The distance the earth travels in one year is one full turn around a circle with radius 9.3 x 107 _{miles. Using the}

formula for circumference, and approximating



≈ 58.4 x 107

≈ 5.84 x 108 _miles

In decimal form, this is 584,000,000, or 584 million miles!

EXAMPLE 8

Applying Scientific Notation to

Astronomy