 # Mathematical Operations Using Numbers in Scientific Notation.

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Mathematical Operations Using Numbers in Scientific Notation

Adding All numbers must be expressed in the same power of 10 (A x 10 m ) + (B x 10 m )  (A + B) x 10 m If exponents are already the same… (2.2 x 10³) + (4.12 x 10 3 ) = 2.2 x 10³ + 4.12 x 10 3 6.32 x 10 3 = 6.3 x 10 3 (round to correct sig figs)

Adding All numbers must be expressed in the same power of 10 (A x 10m) + (B x 10m)  (A + B) x 10m If exponents are already the same… (1.51 x 10‾2) + (9.34 x 10‾2) = 1.51 + 9.34 10.85 x 10‾2 needs to be converted to proper scientific notation or 1.085 x 10‾1 (answer is in correct sig figs)

To put 10.85 x 10‾ 2 in proper scientific notation we need to move the decimal and change the exponent. So : If you move the decimal to the right, add (-1) to the exponent If you move the decimal to the left, add (+1) to the exponent

10.85 X 10 -2 need to move the decimal to the left so will add a (+1) = 10.85 X 10 -2+1 = 1.085 X 10 -1 If your result is: 0.233 x 10 2 need to move the decimal to the right so will add a (-1) = 0.233 x 10 2-1 = 2.33 X 10 1

Different Exponents (1.234 x 10‾³) + (5.623 x 10‾²) = Doesn’t matter which exponent you change (1.234 x 10‾³) + (56.23 x 10 -2+-1=-3 ) = 57.464 x 10‾² 1.234 +56.23 57.464 x 10‾³ = 57.46 x 10‾³ = 5.746 x 10‾² (0.1234 x 10‾²) + (5.623 x 10‾²) = 5.746 x 10‾²

Addition (1.234 x 10‾³) + (5.623 x 10‾²) = Doesn’t matter which exponent you change (0.1234 x 10‾²) + (5.623 x 10‾²) = 5.7464 x 10‾ 2 = 5.746 x 10‾ 2 OR (1.234 x 10‾³) - (56.23 x 10 -2+-1=-3 ) = 57.464 x 10‾² 1.234 - 56.23 -57.464 x 10‾³ = -5.746 x 10‾²

Check your work! (1.234 x 10‾³) + (5.623 x 10‾²) = 0.001234 + 0.05623 = 0.001234 +0.05623 0.057464 = 5.746 x 10‾²

Subtracting 2000 X 10 4 – 5 X 10 4 = 1995 X 10 4 Need to round answer to correct sig figs! 1995 X 10 4 becomes 2000 X 10 4 Still not done! 2000 X 10 4 = move the decimal 3 places to the left and add “3” to the exponent 2000 X 10 4+3 = 2 x 10 7

Multiplying Multiply the decimal parts Add the exponents of 10s (A x 10 m ) x (B x 10 n )  (A x B) x 10 (m +n) (1.23 x 10 3 ) x (7.60 x 10 2 ) = (1.23 x 7.60) x 10 (3 + 2) = 9.348 x 10 5 = 9.35 x 10 5 (ROUND TO CORRECT SIG FIGS)

Example (4.16 x 10 3 )(2 x 10 4 ) =

Dividing Divide the decimal parts Subtract the exponents (A x 10 x )  (B x 10 y )  (A  B) x 10 (x-y) or A B x 10 (x-y)

Example: (4.68 x 10 -3 ) ÷ (4.00 x 10 -5 ) 4.68 4.00 x 10 -3-(-5) = 1.17 x 10 2

Using Pre-determined Measurements in Calculations If a value given is a measurement and is used in a calculation, it will influence the number of sig figs in your answer.

Using Pre-determined Measurements in Calculations Values of gravity could be: 6.7 X 10 -11 N m 2 kg -2 6.672 X 10 -11 N m 2 kg -2 6.67 X 10 -11 N m 2 kg -2  All three values represent the force of gravity on earth but they are expressed with a different degree of accuracy.  Therefore, where this value appears in a calculation, it would influence the number of significant digits used in the final answer.

Classroom exercises