Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent

The Quotient Property of Exponents, introduced in Divide Monomials, had two forms depending on whether the exponent in the numerator or denominator was larger.

What if we just subtract exponents, regardless of which is larger? Let’s consider $\frac{x^{2}}{x^{5}} .$

We subtract the exponent in the denominator from the exponent in the numerator.

\frac{x^{2}}{x^{5}}

x^{2 - 5}

x^{−3}

We can also simplify $\frac{x^{2}}{x^{5}}$ by dividing out common factors: $\frac{x^{2}}{x^{5}} .$

$A fraction is shown. The numerator is x times x, the denominator is x times x times x times x times x. Two x's are crossed out in red on the top and on the bottom. Below that, the fraction 1 over x cubed is shown.$

This implies that $x^{−3} = \frac{1}{x^{3}}$ and it leads us to the definition of a negative exponent.

The negative exponent tells us to re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write an expression with only positive exponents.

Simplify:

ⓐ $4^{−2}$
ⓑ $10^{−3}$

Solution

Steps to simplify an expression with a negative exponent, demonstrating the conversion and calculation of 4^-2 to 1/16.
ⓐ
	$4^{−2}$
Use the definition of a negative exponent, $a^{- n} = \frac{1}{a^{n}} .$	$\frac{1}{4^{2}}$
Simplify.	$\frac{1}{16}$

Step-by-step simplification of 10 to the power of -3, demonstrating the use of negative exponent definition.
ⓑ
	$10^{−3}$
Use the definition of a negative exponent, $a^{- n} = \frac{1}{a^{n}} .$	$\frac{1}{10^{3}}$
Simplify.	$\frac{1}{1000}$

When simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent.

Simplify:

ⓐ ${(−3)}^{−2}$
ⓑ ${−3}^{−2}$

Solution

The negative in the exponent does not affect the sign of the base.

Steps to simplify an expression with a negative exponent, detailing each mathematical operation.
ⓐ
The exponent applies to the base, $- 3$ .	${(−3)}^{−2}$
Take the reciprocal of the base and change the sign of the exponent.	$\frac{1}{{(−3)}^{2}}$
Simplify.	$\frac{1}{9}$

Step-by-step evaluation of -3^(-2), demonstrating rules for negative exponents and bases.
ⓑ
The expression $- 3^{−2}$ means "find the opposite of $3^{−2}$ ". The exponent applies only to the base, 3.	$- 3^{−2}$
Rewrite as a product with −1.	$−1 \cdot 3^{−2}$
Take the reciprocal of the base and change the sign of the exponent.	$−1 \cdot \frac{1}{3^{2}}$
Simplify.	$- \frac{1}{9}$

We must be careful to follow the order of operations. In the next example, parts ⓐ and ⓑ look similar, but we get different results.

Simplify:

ⓐ $4 \cdot 2^{−1}$
ⓑ ${(4 \cdot 2)}^{−1}$

Solution

Remember to always follow the order of operations.

Step-by-step evaluation of 4 * 2^-1, demonstrating exponent rules and order of operations.
ⓐ
Do exponents before multiplication.	$4 \cdot 2^{−1}$
Use $a^{- n} = \frac{1}{a^{n}} .$	$4 \cdot \frac{1}{2^{1}}$
Simplify.	$2$

Demonstrates the step-by-step simplification of (4 * 2)^-1 using negative exponent rules.
ⓑ	${(4 \cdot 2)}^{−1}$
Simplify inside the parentheses first.	${(8)}^{−1}$
Use $a^{- n} = \frac{1}{a^{n}} .$	$\frac{1}{8^{1}}$
Simplify.	$\frac{1}{8}$

When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers.

Simplify: $x^{−6} .$

Solution

Demonstrates how to apply the definition of a negative exponent to simplify a mathematical expression.
	$x^{−6}$
Use the definition of a negative exponent, $a^{- n} = \frac{1}{a^{n}} .$	$\frac{1}{x^{6}}$

When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the order of operations, expressions in parentheses are simplified before exponents are applied. We’ll see how this works in the next example.

Simplify:

ⓐ $5 y^{−1}$
ⓑ ${(5 y)}^{−1}$
ⓒ ${(−5 y)}^{−1}$

Solution

This table illustrates the step-by-step simplification of the expression 5y^-1, demonstrating how to handle negative exponents.
ⓐ
Notice the exponent applies to just the base $y$ .	$5 y^{−1}$
Take the reciprocal of $y$ and change the sign of the exponent.	$5 \cdot \frac{1}{y^{1}}$
Simplify.	$\frac{5}{y}$

Steps demonstrating the simplification of the algebraic expression (5y)^-1 using rules for negative exponents.
ⓑ
Here the parentheses make the exponent apply to the base $5 y$ .	${(5 y)}^{−1}$
Take the reciprocal of $5 y$ and change the sign of the exponent.	$\frac{1}{{(5 y)}^{1}}$
Simplify.	$\frac{1}{5 y}$

Step-by-step simplification of the algebraic expression (-5y)^-1, demonstrating the rule for negative exponents.
ⓒ
	${(−5 y)}^{−1}$
The base is $- 5 y$ . Take the reciprocal of $- 5 y$ and change the sign of the exponent.	$\frac{1}{{(−5 y)}^{1}}$
Simplify.	$\frac{1}{−5 y}$
Use $\frac{a}{- b} = - \frac{a}{b} .$	$- \frac{1}{5 y}$

Now that we have defined negative exponents, the Quotient Property of Exponents needs only one form, $\frac{a^{m}}{a^{n}} = a^{m - n},$ where $a \neq 0$ and m and n are integers.

When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. If the result gives us a negative exponent, we will rewrite it by using the definition of negative exponents, $a^{- n} = \frac{1}{a^{n}} .$

Simplify Expressions with Integer Exponents

All the exponent properties we developed earlier in this chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.

Simplify:

ⓐ $x^{−4} \cdot x^{6}$
ⓑ $y^{−6} \cdot y^{4}$
ⓒ $z^{−5} \cdot z^{−3}$

Solution

Steps for simplifying the exponential expression x^(-4) * x^6 using the product property, resulting in x^2.
ⓐ
	$x^{−4} \cdot x^{6}$
Use the Product Property, $a^{m} \cdot a^{n} = a^{m + n} .$	$x^{−4 + 6}$
Simplify.	$x^{2}$

Step-by-step simplification of an exponential expression involving the product rule and negative exponents.
ⓑ
	$y^{−6} \cdot y^{4}$
The bases are the same, so add the exponents.	$y^{−6 + 4}$
Simplify.	$y^{−2}$
Use the definition of a negative exponent, $a^{- n} = \frac{1}{a^{n}} .$	$\frac{1}{y^{2}}$

Steps for simplifying the exponential expression z^(-5) * z^(-3) by applying the product rule and negative exponent definition.
ⓒ
	$z^{−5} \cdot z^{−3}$
The bases are the same, so add the exponents.	$z^{−5−3}$
Simplify.	$z^{−8}$
Use the definition of a negative exponent, $a^{- n} = \frac{1}{a^{n}} .$	$\frac{1}{z^{8}}$

In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property of Exponents.

Simplify: $(m^{4} n^{−3}) (m^{−5} n^{−2}) .$

Solution

Demonstrates the step-by-step simplification of an algebraic expression involving exponents.
	$(m^{4} n^{−3}) (m^{−5} n^{−2})$
Use the Commutative Property to get like bases together.	$m^{4} m^{−5} \cdot n^{−2} n^{−3}$
Add the exponents for each base.	$m^{−1} \cdot n^{−5}$
Take reciprocals and change the signs of the exponents.	$\frac{1}{m^{1}} \cdot \frac{1}{n^{5}}$
Simplify.	$\frac{1}{m n^{5}}$

If the monomials have numerical coefficients, we multiply the coefficients, just as we did in Use Multiplication Properties of Exponents.

Simplify: $(2 x^{−6} y^{8}) (−5 x^{5} y^{−3}) .$

Solution

Step-by-step simplification of an algebraic expression involving exponents.
	$(2 x^{−6} y^{8}) (−5 x^{5} y^{−3})$
Rewrite with the like bases together.	$2 (−5) \cdot (x^{−6} x^{5}) \cdot (y^{8} y^{−3})$
Simplify.	$−10 \cdot x^{−1} \cdot y^{5}$
Use the definition of a negative exponent, $a^{- n} = \frac{1}{a^{n}} .$	$−10 \cdot \frac{1}{x^{1}} \cdot y^{5}$
Simplify.	$\frac{−10 y^{5}}{x}$

In the next two examples, we’ll use the Power Property and the Product to a Power Property.

Simplify: ${(k^{3})}^{−2} .$

Solution

Illustrates the step-by-step simplification of the exponential expression (k^3)^(-2) using exponent rules, yielding 1/k^6.
	${(k^{3})}^{−2}$
Use the Product to a Power Property, ${(a b)}^{m} = a^{m} b^{m} .$	$k^{3 (−2)}$
Simplify.	$k^{−6}$
Rewrite with a positive exponent.	$\frac{1}{k^{6}}$

Simplify: ${(5 x^{−3})}^{2} .$

Solution

Step-by-step simplification of the algebraic expression (5x^-3)^2 using exponent properties.
	${(5 x^{−3})}^{2}$
Use the Product to a Power Property, ${(a b)}^{m} = a^{m} b^{m} .$	$5^{2} {(x^{−3})}^{2}$
Simplify $5^{2}$ and multiply the exponents of $x$ using the Power Property, ${(a^{m})}^{n} = a^{m \cdot n} .$	$25 x^{−6}$
Rewrite $x^{−6}$ by using the definition of a negative exponent, $a^{- n} = \frac{1}{a^{n}} .$	$25 \cdot \frac{1}{x^{6}}$
Simplify	$\frac{25}{x^{6}}$

To simplify a fraction, we use the Quotient Property.

Simplify: $\frac{r^{5}}{r^{−4}} .$

Solution


Use the Quotient Property, $\frac{a^{m}}{a^{n}} = a^{m - n}$ .

Simplify.

Convert from Decimal Notation to Scientific Notation

Remember working with place value for whole numbers and decimals? Our number system is based on powers of $10 .$ We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on.

Consider the numbers $4000$ and $0.004 .$ We know that $4000$ means $4 \times 1000$ and $0.004$ means $4 \times \frac{1}{1000} .$ If we write the $1000$ as a power of ten in exponential form, we can rewrite these numbers in this way:

\begin{matrix} 4000 & 0.004 \\ 4 \times 1000 & 4 \times \frac{1}{1000} \\ 4 \times 10^{3} & 4 \times \frac{1}{10^{3}} \\ 4 \times 10^{−3} \end{matrix}

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than $10,$ and the second factor is a power of $10$ written in exponential form, it is said to be in scientific notation.

It is customary in scientific notation to use $\times$ as the multiplication sign, even though we avoid using this sign elsewhere in algebra.

Scientific notation is a useful way of writing very large or very small numbers. It is used often in the sciences to make calculations easier.

If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.

On the left, we see 4000 equals 4 times 10 cubed. Beneath that is the same thing, but there is an arrow from after the last 0 in 4000 to between the 4 and the first 0. Beneath, it says, “Moved the decimal point 3 places to the left.” On the right, we see 0.004 equals 4 times 10 to the negative 3. Beneath that is the same thing, but there is an arrow from the decimal point to after the 4. Beneath, it says, “Moved the decimal point 3 places to the right.”

In both cases, the decimal was moved $3$ places to get the first factor, $4,$ by itself.

The power of $10$ is positive when the number is larger than $1 : 4000 = 4 \times 10^{3} .$
The power of $10$ is negative when the number is between $0$ and $1 : 0.004 = 4 \times 10^{- 3} .$

Write $37,000$ in scientific notation.

Solution

Step 1: Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
Step 2: Count the number of decimal places, $n$ , that the decimal point was moved.	3.70000 4 places
Step 3: Write the number as a product with a power of 10.	$3.7 \times 10^{4}$
If the original number is: greater than 1, the power of 10 will be $10^{n}$ . between 0 and 1, the power of 10 will be $10^{−n}$
Step 4: Check.
$10^{4}$ is 10,000 and 10,000 times 3.7 will be 37,000.
	$37,000 = 3.7 \times 10^{4}$

Write in scientific notation: $0.0052 .$

Solution

	0.0052
Move the decimal point to get 5.2, a number between 1 and 10.
Count the number of decimal places the point was moved.	3 places
Write as a product with a power of 10.	$5.2 \times 10^{−3}$
Check your answer: $\begin{matrix} 5.2 \times 10^{−3} \\ 5.2 \times \frac{1}{10^{3}} \\ 5.2 \times \frac{1}{1000} \\ 5.2 \times 0.001 \\ 0.0052 \end{matrix}$
	$0.0052 = 5.2 \times 10^{−3}$

Convert Scientific Notation to Decimal Form

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.

\begin{matrix} 9.12 \times 10^{4} & 9.12 \times 10^{−4} \\ 9.12 \times 10,000 & 9.12 \times 0.0001 \\ 91,200 & 0.000912 \end{matrix}

If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.

On the left, we see 9.12 times 10 to the 4th equals 91,200. Beneath that is 9.12 followed by 2 spaces, with an arrow from the decimal to after the second space, times 10 to the 4th equals 91,200. On the right, we see 9.12 times 10 to the negative 4 equals 0.000912. Beneath that is three spaces followed by 9.12 with an arrow from the decimal to after the first space, times 10 to the negative 4 equals 0.000912.

In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.

Convert to decimal form: $6.2 \times 10^{3} .$

Solution

Step 1: Determine the exponent, $n$ , on the factor 10.	$6.2 \times 10^{3}$
Step 2: Move the decimal point $n$ places, adding zeros if needed.
If the exponent is positive, move the decimal point $n$ places to the right. If the exponent is negative, move the decimal point $\| n \|$ places to the left.	6,200
Step 3: Check to see if your answer makes sense.
$10^{3}$ is 1000 and 1000 times 6.2 will be 6,200.	$6.2 \times 10^{3} = 6,200$

Convert to decimal form: $8.9 \times 10^{−2} .$

Solution

	$8.9 \times 10^{−2}$
Determine the exponent $n$ , on the factor 10.	The exponent is −2.
Move the decimal point 2 places to the left.
Add zeros as needed for placeholders.	0.089
	$8.9 \times 10^{−2} = 0.089$
The Check is left to you.

Multiply and Divide Using Scientific Notation

We use the Properties of Exponents to multiply and divide numbers in scientific notation.

Multiply. Write answers in decimal form: $(4 \times 10^{5}) (2 \times 10^{−7}) .$

Solution

Steps to multiply numbers in scientific notation, showing rearrangement, multiplication, and conversion to decimal form.
	$(4 \times 10^{5}) (2 \times 10^{−7})$
Use the Commutative Property to rearrange the factors.	$4 \cdot 2 \cdot 10^{5} \cdot 10^{−7}$
Multiply 4 by 2 and use the Product Property to multiply $10^{5}$ by $10^{−7}$ .	$8 \times 10^{−2}$
Change to decimal form by moving the decimal two places left.	$0.08$

Divide. Write answers in decimal form: $\frac{9 \times 10^{3}}{3 \times 10^{−2}} .$

Solution

Steps for simplifying a fraction with scientific notation, demonstrating the process from separation of factors to final decimal form.
	$\frac{9 \times 10^{3}}{3 \times 10^{−2}}$
Separate the factors.	$\frac{9}{3} \times \frac{10^{3}}{10^{−2}}$
Divide 9 by 3 and use the Quotient Property to divide $10^{3}$ by $10^{−2}$ .	$3 \times 10^{5}$
Change to decimal form by moving the decimal five places right.	$300,000$

Key Concepts

Summary of Exponent Properties
- If $a, b$ are real numbers and $m, n$ are integers, then
  $\begin{matrix} Product Property & a^{m} \cdot a^{n} = a^{m + n} \\ Power Property & {(a^{m})}^{n} = a^{m \cdot n} \\ Product to a Power Property & {(a b)}^{m} = a^{m} b^{m} \\ Quotient Property & \frac{a^{m}}{a^{n}} = a^{m - n}, a \neq 0 \\ Zero Exponent Property & a^{0} = 1, a \neq 0 \\ Quotient to a Power Property & {(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}, b \neq 0 \\ Definition of Negative Exponent & a^{- n} = \frac{1}{a^{n}} \end{matrix}$
Convert from Decimal Notation to Scientific Notation: To convert a decimal to scientific notation:
1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
2. Count the number of decimal places, $n$ , that the decimal point was moved.
3. Write the number as a product with a power of 10.
  - If the original number is greater than 1, the power of 10 will be $10^{n}$ .
  - If the original number is between 0 and 1, the power of 10 will be $10^{- n}$ .
4. Check.
Convert Scientific Notation to Decimal Form: To convert scientific notation to decimal form:
1. Determine the exponent, $n$ , on the factor 10.
2. Move the decimal n places, adding zeros if needed.
  - If the exponent is positive, move the decimal point $n$ places to the right.
  - If the exponent is negative, move the decimal point $| n |$ places to the left.
3. Check.

Practice Makes Perfect

Use the Definition of a Negative Exponent

In the following exercises, simplify.

$5^{−3}$

$8^{−2}$

Solution

$\frac{1}{64}$

$3^{−4}$

$2^{−5}$

Solution

$\frac{1}{32}$

$7^{−1}$

$10^{−1}$

Solution

$\frac{1}{10}$

$2^{−3} + 2^{−2}$

$3^{−2} + 3^{−1}$

Solution

$\frac{4}{9}$

$3^{−1} + 4^{−1}$

$10^{−1} + 2^{−1}$

Solution

$\frac{3}{5}$

$10^{0} - 10^{−1} + 10^{−2}$

$2^{0} - 2^{−1} + 2^{−2}$

Solution

$\frac{3}{4}$

ⓐ ${(−6)}^{−2}$
ⓑ $- 6^{−2}$

ⓐ ${(−8)}^{−2}$
ⓑ $- 8^{−2}$

Solution

ⓐ $\frac{1}{64}$
ⓑ $- \frac{1}{64}$

ⓐ ${(−10)}^{−4}$
ⓑ $- 10^{−4}$

ⓐ ${(−4)}^{−6}$
ⓑ $- 4^{−6}$

Solution

ⓐ $\frac{1}{4096}$
ⓑ $- \frac{1}{4096}$

ⓐ $5 \cdot 2^{−1}$
ⓑ ${(5 \cdot 2)}^{−1}$

ⓐ $10 \cdot 3^{−1}$
ⓑ ${(10 \cdot 3)}^{−1}$

Solution

ⓐ $\frac{10}{3}$
ⓑ $\frac{1}{30}$

ⓐ $4 \cdot 10^{−3}$
ⓑ ${(4 \cdot 10)}^{−3}$

ⓐ $3 \cdot 5^{−2}$
ⓑ ${(3 \cdot 5)}^{−2}$

Solution

ⓐ $\frac{3}{25}$
ⓑ $\frac{1}{225}$

$n^{−4}$

$p^{−3}$

Solution

$\frac{1}{p^{3}}$

$c^{−10}$

$m^{−5}$

Solution

$\frac{1}{m^{5}}$

ⓐ $4 x^{−1}$
ⓑ ${(4 x)}^{−1}$
ⓒ ${(−4 x)}^{−1}$

ⓐ $3 q^{−1}$
ⓑ ${(3 q)}^{−1}$
ⓒ ${(−3 q)}^{−1}$

Solution

ⓐ $\frac{3}{q}$
ⓑ $\frac{1}{3 q}$
ⓒ $- \frac{1}{3 q}$

ⓐ $6 m^{−1}$
ⓑ ${(6 m)}^{−1}$
ⓒ ${(−6 m)}^{−1}$

ⓐ $10 k^{−1}$
ⓑ ${(10 k)}^{−1}$
ⓒ ${(−10 k)}^{−1}$

Solution

ⓐ $\frac{10}{k}$
ⓑ $\frac{1}{10 k}$
ⓒ $- \frac{1}{10 k}$

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

$p^{−4} \cdot p^{8}$

$r^{−2} \cdot r^{5}$

Solution

r³

$n^{−10} \cdot n^{2}$

$q^{−8} \cdot q^{3}$

Solution

$\frac{1}{q^{5}}$

$k^{−3} \cdot k^{−2}$

$z^{−6} \cdot z^{−2}$

Solution

$\frac{1}{z^{8}}$

$a \cdot a^{−4}$

$m \cdot m^{−2}$

Solution

$\frac{1}{m}$

$p^{5} \cdot p^{−2} \cdot p^{−4}$

$x^{4} \cdot x^{−2} \cdot x^{−3}$

Solution

$\frac{1}{x}$

$a^{3} b^{−3}$

$u^{2} v^{−2}$

Solution

$\frac{u^{2}}{v^{2}}$

$(x^{5} y^{−1}) (x^{−10} y^{−3})$

$(a^{3} b^{−3}) (a^{−5} b^{−1})$

Solution

$\frac{1}{a^{2} b^{4}}$

$(u v^{−2}) (u^{−5} v^{−4})$

$(p q^{−4}) (p^{−6} q^{−3})$

Solution

$\frac{1}{p^{5} q^{7}}$

$(−2 r^{−3} s^{9}) (6 r^{4} s^{−5})$

$(−3 p^{−5} q^{8}) (7 p^{2} q^{−3})$

Solution

$- \frac{21 q^{5}}{p^{3}}$

$(−6 m^{−8} n^{−5}) (−9 m^{4} n^{2})$

$(−8 a^{−5} b^{−4}) (−4 a^{2} b^{3})$

Solution

$\frac{32}{a^{3} b}$

${(a^{3})}^{−3}$

${(q^{10})}^{−10}$

Solution

$\frac{1}{q^{100}}$

${(n^{2})}^{−1}$

${(x^{4})}^{−1}$

Solution

$\frac{1}{x^{4}}$

${(y^{−5})}^{4}$

${(p^{−3})}^{2}$

Solution

$\frac{1}{p^{6}}$

${(q^{−5})}^{−2}$

${(m^{−2})}^{−3}$

Solution

m⁶

${(4 y^{−3})}^{2}$

${(3 q^{−5})}^{2}$

Solution

$\frac{9}{q^{10}}$

${(10 p^{−2})}^{−5}$

${(2 n^{−3})}^{−6}$

Solution

$\frac{n^{18}}{64}$

$\frac{u^{9}}{u^{−2}}$

$\frac{b^{5}}{b^{−3}}$

Solution

b⁸

$\frac{x^{−6}}{x^{4}}$

$\frac{m^{5}}{m^{−2}}$

Solution

m⁷

$\frac{q^{3}}{q^{12}}$

$\frac{r^{6}}{r^{9}}$

Solution

$\frac{1}{r^{3}}$

$\frac{n^{−4}}{n^{−10}}$

$\frac{p^{−3}}{p^{−6}}$

Solution

p³

Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

45,000

280,000

Solution

2.8 × 10⁵

8,750,000

1,290,000

Solution

1.29 × 10⁶

0.036

0.041

Solution

4.1 × 10⁻²

0.00000924

0.0000103

Solution

1.03 × 10⁻⁵

The population of the United States on July 4, 2010 was almost $310,000,000 .$

The population of the world on July 4, 2010 was more than $6,850,000,000 .$

Solution

6.85 × 10⁹

The average width of a human hair is $0.0018$ centimeters.

The probability of winning the $2010$ Megamillions lottery is about $0.0000000057 .$

Solution

5.7 × 10⁻⁹

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

$4.1 \times 10^{2}$

$8.3 \times 10^{2}$

Solution

830

$5.5 \times 10^{8}$

$1.6 \times 10^{10}$

Solution

16,000,000,000

$3.5 \times 10^{−2}$

$2.8 \times 10^{−2}$

Solution

0.028

$1.93 \times 10^{−5}$

$6.15 \times 10^{−8}$

Solution

0.0000000615

In 2010, the number of Facebook users each day who changed their status to ‘engaged’ was $2 \times 10^{4} .$

At the start of 2012, the US federal budget had a deficit of more than $$1.5 \times 10^{13} .$

Solution

$15,000,000,000,000

The concentration of carbon dioxide in the atmosphere is $3.9 \times 10^{−4} .$

The width of a proton is $1 \times 10^{−5}$ of the width of an atom.

Solution

0.00001

Multiply and Divide Using Scientific Notation

In the following exercises, multiply or divide and write your answer in decimal form.

$(2 \times 10^{5}) (2 \times 10^{−9})$

$(3 \times 10^{2}) (1 \times 10^{−5})$

Solution

0.003

$(1.6 \times 10^{−2}) (5.2 \times 10^{−6})$

$(2.1 \times 10^{−4}) (3.5 \times 10^{−2})$

Solution

0.00000735

$\frac{6 \times 10^{4}}{3 \times 10^{−2}}$

$\frac{8 \times 10^{6}}{4 \times 10^{−1}}$

Solution

20,000,000

$\frac{7 \times 10^{−2}}{1 \times 10^{−8}}$

$\frac{5 \times 10^{−3}}{1 \times 10^{−10}}$

Solution

50,000,000

Everyday Math

Calories In May 2010 the Food and Beverage Manufacturers pledged to reduce their products by $1.5$ trillion calories by the end of 2015.

ⓐ Write $1.5$ trillion in decimal notation.
ⓑ Write $1.5$ trillion in scientific notation.

Length of a year The difference between the calendar year and the astronomical year is $0.000125$ day.

ⓐ Write this number in scientific notation.
ⓑ How many years does it take for the difference to become 1 day?

Solution

ⓐ 1.25 × 10⁻⁴
ⓐ 8,000

Calculator display Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the probability of getting a particular 5-card hand from a deck of cards, Mario divided $1$ by $2,598,960$ and saw the answer $3.848 \times 10^{−7} .$ Write the number in decimal notation.

Calculator display Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the number of ways Barbara could make a collage with $6$ of her $50$ favorite photographs, she multiplied $50 \cdot 49 \cdot 48 \cdot 47 \cdot 46 \cdot 45 .$ Her calculator gave the answer $1.1441304 \times 10^{10} .$ Write the number in decimal notation.

Solution

11,441,304,000

Writing Exercises

ⓐ Explain the meaning of the exponent in the expression $2^{3} .$
ⓑ Explain the meaning of the exponent in the expression $2^{−3}$

When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

Solution

Answers will vary.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

A self-assessment checklist for math skills, including exponents and scientific notation, with options to mark 'Confidently,' 'With some help,' or 'No-I don't get it!'

ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?