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1.2 Scientific Notation and Metric Conversions

1.2.1 Prep

An exponential expression looks like:

[latex]a^n[/latex]

Where a: base and n: exponent (or power)

A positive integer exponent tells us how many times to multiply by the base number

For example,

[latex]10^3=10\cdot10\cdot10\\\;\;\;\;\;\;=1,000[/latex]

Write out each exponential expression using multiplication and evaluate.

  1. [latex]10^1[/latex]
  2. [latex]10^2[/latex]
  3. [latex]10^4[/latex]
  4. [latex]10^7[/latex]
  5. Discuss what happens when you raise 10 to a positive integer power.

Can you write the following numbers as powers of 10?

  1. 10,000
  2. 100,000,000

A negative integer exponent tells us how many times to divide by the base number.

Remember! Division can also be represented with a fraction bar.

For example,

[latex]\begin{array}{l}10^{-3}=1\div10\div10\div10\\=\frac1{10}⋅\frac1{10}⋅\frac1{10}\\=0.001\operatorname{or}\frac1{1,000}\end{array}[/latex]

Write out each exponential expression using division and evaluate.

  1. [latex]10^{-1}[/latex]
  2. [latex]10^{-2}[/latex]
  3. [latex]10^{-4}[/latex]
  4. [latex]10^{-7}[/latex]
  5. Discuss what happens when you raise 10 to a negative integer power.

Can you write the following numbers as powers of 10?

  1. 0.00001
  2. 0.000000001

In mathematical order of operations, exponents come before multiplication or division. So for:

[latex]5\times10^3[/latex]

You have to evaluate the exponent of 3 before multiplying by the 5.

[latex]5\times1,000[/latex]

[latex]5,000[/latex]

Also keep in mind that multiplication is not always as obvious as the “[latex]\times[/latex]” symbol. The above could also be written as:

[latex]5(10)^3 \text{ or } 5\cdot10^3[/latex]

And it might look more tempting to start by multiplying the 5 and the 10. But resist this urge — start with the exponent!

Follow correct order of operations and evaluate the following.

  1. [latex]7\times10^2[/latex]
  2. [latex]3\times10^{-4}[/latex]
  3. [latex]7.45\times10^6[/latex]
  4. [latex]3.14\times10^{-6}[/latex]

1.2.2 Preview

Just as multiplication is a shorthand for addition, a positive integer exponent is a shorthand for repeated multiplication of the same number.

[latex]10\times10=10^2[/latex]

Try It!

Write [latex]10\times10\times10\times10\times10[/latex] using an exponent

The following numbers are written using powers of 10.

[latex]\begin{array}{lcccc}2&=&2&=&2\times10^0\\20&=&2\times10&=&2\times10^1\\200&=&2\times10\times10&=&2\times10^2\\2,000&=&2\times10\times10\times10&=&2\times10^3\end{array}[/latex]

Try It!

Write 2, 000, 000 using the pattern above.

Scientific Notation

[latex]\left(\text{a number no less than 1 but less than 10}\right)\times10^{\left(\text{exponent}\right)}[/latex]

Try It!

Convert from scientific to decimal notation.

[latex]7.3\times10^4[/latex]

A negative integer exponent is a shorthand for repeated division of the same number.

 

[latex]\begin{array}{lr}1\div10=\frac1{10}&=10^{-1}\\1\div10\div10=\frac1{10\times10}&=10^{-2}\\1\div10\div10\div10=\frac1{10\times10\times10}&=10^{-3}\end{array}[/latex]

Try It!

Write [latex]1\div10\div10\div10\div10\div10[/latex]

The following numbers are written using powers of 10.

[latex]\begin{array}{lcc}0.5&=&5\times10^{-1}\\0.05&=&5\times10^{-2}\\0.005&=&5\times10^{-3}\end{array}[/latex]

Try It!

Convert from scientific to decimal notation.

\[4.2\times10^{-5}\]

Search the web to find real-life examples of the following measurements, and compare your answers with classmates.

Measurement Example
1 m The length of a guitar
1 cm
1 mm
1 hm The length of a football field
1 km
1 inch
1 yard
1 mile
1 g The mass of a paperclip
1 kg
[latex]1\mu g[/latex] Less than a grain of sand
1 L
1 mL 20 drops of water
1 quart
1 fluid ounce
[latex]1cm^2[/latex]
[latex]1m^2[/latex] The area covered by a large opened umbrella
1 acre

1.2.3 Class Examples

Scientific Notation

Convert from decimal to scientific notation.

  1. 35,400,000
  2. 0.000000009876

Convert from scientific to decimal notation.

  1. [latex]2.32\times10^6[/latex]
  2. [latex]5.2\times10^{-3}[/latex]

Metric Measurements

Fill in the table real-life examples of the following measurements, and add more answers from your classmates.

Measurement Example
1 m
1 cm
1 mm
1 hm
1 g
1 kg
[latex]1\mu g[/latex]
1 L
1 mL
[latex]1cm^2[/latex]
[latex]1m^2[/latex]

Metric Prefixes

Prefix Mega- Kilo- Hecto- Deka- Unit Deci- Centi- Milli- Micro-
Abbreviation M k h da m, g, L d c m [latex]\mu[/latex]
No. of units 1,000,000 1,000 100 10 1 [latex]\frac1{10}[/latex] [latex]\frac1{100}[/latex] [latex]\frac1{1000}[/latex] [latex]\frac1{1000000}[/latex]
Power of 10 [latex]10^6[/latex] [latex]10^3[/latex] [latex]10^2[/latex] [latex]10^1[/latex] [latex]10^0[/latex] [latex]10^{-1}[/latex] [latex]10^{-2}[/latex] [latex]10^{-3}[/latex] [latex]10^{-6}[/latex]

Metric System Conversions

Convert

  1. 50 cm = ______________ m
  2. 8 kg = ______________ g
  3. 9.1 kL = ______________ L
  4. 0.098 g= ______________ mg
  5. 9.1 mL = ______________ L
  6. 0.72 km = ______________ mm
  7. 0.568 kg = ______________ mg
  8. 0.098 g = ______________ [latex]\mu g[/latex]
  9. 2000 mg = ______________ [latex]\mu g[/latex]
  10. [latex]4\;m^2[/latex] = ______________ [latex]cm^2[/latex]
  11. [latex]0.00025\;hm^3[/latex] = ______________ [latex]cm^3[/latex]
  12. England is 130.4 billion square meters. Convert this to square kilometers.

1.2.4 Homework

Use a calculator to simplify. Write answers in scientific notation rounded to three digits.

  1. [latex]-1.53\times10^3+4.57\times10^3[/latex]
  2. [latex]1.25\times10^{-6}-4.79\times10^{-3}[/latex]
  3. [latex]\left(1.24\times10^3\right)\left(5.79\times10^{-9}\right)[/latex]
  4. [latex]\frac{5.47\times10^3}{1.76\times10^{-3}}[/latex]
  5. Answer the following.
    1. Find an example of a number in the real world that is written in scientific notation. (Be sure to include its context and its units.)
    2. Write this number in decimal notation.
    3. Explain when it might be advantageous to write this number in scientific notation.

Perform the following metric conversions, and write your answer in scientific notation.

  1. A nurse needs to give a patient 0.0025 grams of Valium.
    1. Convert this dosage to scientific notation.
    2. Convert this dosage to mg.
    3. Convert this dosage to [latex]\mu g[/latex]
    4. Which way of expressing this dosage is most appropriate, and why?

Convert. Then write your answer in scientific notation.

  1. 0.5 L = ________________ mL
  2. 16cm = ________________ mm
  3. 104 km = ________________ cm
  4. [latex]0.048\;m^2[/latex] = ________________ [latex]cm^2[/latex]
  5. 1.98 g = ________________ kg
  6. 2500 cm = ________________ km
  7. [latex]160\;cm^3[/latex] = ________________ [latex]mm^3[/latex]
  8. 12 L = ________________ ML
  9. Spain is [latex]504,646\;km^2[/latex]. France is [latex]640,679,000,000\;m^2[/latex]. Which country is larger and by how much?

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College Algebra for Non-STEM Majors Copyright © by Amy Collins Montalbano. All Rights Reserved.

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