A set of intermediate algebra problems, related to scientific notation, with answers, are presented. The solutions are at the bottom of the page.
Review: A number x is written in scientific notation when it is written in the form:
x = a × 10^{ N}
with 1 ≤ a < 10 and N is an integer.

Convert from decimal notation to scientific notation.
a) 234
b) 0.000052
c) 220,000
d) 0.00000000000000182

Convert from scientific to decimal notation.
a) 1.01 × 10^{ 2}
b) 8.2 × 10^{ 6}
c) 3.456 × 10^{ 10}
d) 1.2 × 10^{ 8}

Perform the multiplication and write the result in scientific notation.
a) (5 × 10^{ 2})(7 × 10^{ 5})
b) (1.2 × 10^{ 6})(3.5 × 10^{ 14})
c) (4.5 × 10^{ 7})(8.62 × 10^{ 10})
d) (7.2 × 10^{ 6})(0.00039)

Perform the division and write the result in scientific notation.
a) (4 × 10^{ 13}) / (5 × 10^{ 4})
b) 0.000012 / (2.4 × 10^{ 8})
c) (1.2 × 10^{ 6}) / (1.5 × 10^{ 10})
d) 0.00000065 / 0.000013

Evaluate and express in scientific notation.
a) (1.3 × 10^{ 6})^{ 2}
b) (1.5 × 10^{ 8})^{ 2}(2.1 × 10^{ 5})
c) [ (1.144 × 10^{ 2})(6 × 10^{ 7}) ] / (2.2 × 10^{ 4})
d) [ (0.00001)(2,500) ] / [ (0.0003)(150,000) ]

Evaluate in scientific notation.
a) The diameter of an atom: 0.0000000000001 meter
b) diameter of earth = 12,756,000 meters
c) 1 billion: 1000,0000,000
d) 1 micron: 0.000001 meter
e) 1 trillion: 1000,000,000,000
f) Diameter of earth / diameter of atom
g) Speed of light: 300,000,000 m/sec
h) Distance travelled by light in one year.
Answers to the Above Questions

a) 234 = 2.34× 10^{ 2}
b) 0.000052 = 5.2× 10^{ 5}
c) 220,000 = 2.2× 10^{ 5}
d) 0.00000000000000182 = 1.82× 10^{ 15}

a) 1.01 × 10^{ 2} = 101
b) 8.2 × 10^{ 6} = 0.0000082
c) 3.456 × 10^{ 10} = 34560000000
d) 1.2 × 10^{ 8} = 0.000000012

a) (5 × 10^{ 2})(7 × 10^{ 5})
= (5 × 7)(10^{ 2} × 10^{ 5})
= 35 × 10^{ 7} = 3.5 × 10^{ 8}
b) (1.2 × 10^{ 6})(3.5 × 10^{ 14})
= (1.2×3.5)(10^{ 6} × 10^{ 14})
= 4.2 × 10^{ 8}
c) ( 4.5 × 10^{ 7})(8.62 × 10^{ 10})
= ( 4.5 × 8.62)(10^{ 7} × 10^{ 10})
=  38.79 × 10^{ 3} =  3.879 × 10^{ 4}
d) ( 7.2 × 10^{ 6})(0.00039)
= ( 7.2 × 10^{ 6})(3.9 × 10^{ 4})
= ( 7.2 × 3.9)(10^{ 6} × 10^{ 4})
=  28.08 × 10^{ 10} =  2.808 × 10^{ 9}

a) (4 × 10^{ 13}) / (5 × 10^{ 4}) = (4/5)(10^{ 13} / 10^{ 4}) = 0.8 × 10^{ 17} = 8.0 × 10^{ 16}
b) 0.000012 / (2.4 × 10^{ 8}) = (0.000012 / 2.4)(1 / 10^{ 8}) = (0.000005)(1 / 10^{ 8})
= (5.0 × 10^{ 6})(10^{ 8}) = 5.0 × 10^{ 2}
c) (1.2 × 10^{ 6}) / (1.5 × 10^{ 10}) = (1.2 / 1.5)(10^{ 6} / 10^{ 10})
= 0.8 × 10^{ 4} = 8.0 × 10^{ 3}
d) 0.00000065 / 0.000013 = 0.05 = 5.0 × 10^{ 2}

a) (1.3 × 10^{ 6})^{ 2} = (1.3)^{ 2} (10^{ 6})^{ 2} = 1.69 × 10^{ 12}
b) (1.5 × 10^{ 8})^{ 2}(2.1 × 10^{ 5}) = 1.5^{ 2} (10^{ 8})^{ 2} (2.1 × 10^{ 5})
= (2.25 × 10^{ 16})(2.1 × 10^{ 5}) = 4.725 × 10^{ 11}
c) [ (1.144 × 10^{ 2})(6 × 10^{ 7}) ] / (2.2 × 10^{ 4}) = (6.864 × 10^{ 9}) / (2.2 × 10^{ 4})
= (6.864 / 2.2) (10^{ 9} / 10^{ 4}) = 3.12 × 10^{ 5}
d) [ (0.00001)(2,500) ] / [ (0.0003)(150,000) ] = 0.025 / 45 = 0.000555 = 5.6 × 10^{ 4}

a) The diameter of an atom: 0.0000000001 meter = 1.0 × 10^{ 9} meter
b) diameter of earth = 12,756,000 meters = 1.2756 × 10^{ 7} meters
c) 1 billion: 1000,0000,000 = 1.0 × 10^{ 9}
d) 1 micron: 0.000001 meter = 1.0 × 10^{ 6}
e) 1 trillion: 1000,000,000,000 = 1.0 × 10^{ 12}
f) Diameter of earth / diameter of atom = 1.2756 × 10^{ 7} / 1.0 × 10^{ 9} = 1.2756 × 10^{ 16}
g) Speed of light: 300,000,000 meters per second = 3.0 10^{ 8} m/s
h) Distance travelled by light in one year:
There are (365 * 24 * 60 * 60) = 3.1536 × 10^{ 7} seconds in one year.
Distance travelled by light in one year = time * speed of light
= (3.1536 × 10^{ 7})(3.0 10^{ 8}) = 9.4608 × 10^{ 15} meters = 9.4608 × 10^{ 12} kilometers
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