10.5 Integer exponents and scientific notation (Page 4/7)

Prealgebra

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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:

$\begin{matrix} 5.2 \times 10^{−3} \\ 5.2 \times \frac{1}{10^{3}} \\ 5.2 \times \frac{1}{1000} \\ 5.2 \times 0.001 \\ 0.0052 \\ 0.0052 = 5.2 \times 10^{−3} \end{matrix}$

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Write in scientific notation: $0.0078 .$

7.8 × 10 ⁻³

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Write in scientific notation: $0.0129 .$

1.29 × 10 ⁻²

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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.	$6 \times 10^{3}$
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. $6.2 \times 10^{3} = 6,200$

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Convert to decimal form: $1.3 \times 10^{3} .$

1,300

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Convert to decimal form: $9.25 \times 10^{4} .$

92,500

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Convert scientific notation to decimal form.

Determine the exponent, $n,$ on the factor $10 .$
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.
Check.

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.

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Convert to decimal form: $1.2 \times 10^{−4} .$

0.00012

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Convert to decimal form: $7.5 \times 10^{−2} .$

0.075

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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

	$(4 \times 10^{5}) (2 \times 10^{−7})$
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.	$0.08$

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Multiply. Write answers in decimal form: $(3 \times 10^{6}) (2 \times 10^{−8}) .$

0.06

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Multiply. Write answers in decimal form: $(3 \times 10^{−2}) (3 \times 10^{−1}) .$

0.009

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

Solution

	$\frac{9 \times 10^{3}}{3 \times 10^{−2}}$
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.	$300,000$

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Divide. Write answers in decimal form: $\frac{8 \times 10^{4}}{2 \times 10^{−1}} .$

400,000

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Divide. Write answers in decimal form: $\frac{8 \times 10^{2}}{4 \times 10^{−2}} .$

20,000

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Questions & Answers

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Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).

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Source: OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9

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