MATH WORKSHOP
A. Scientific Notation •
•
Put everything into scientific notation before doing any calculations Manipulate the numbers using scientific notation to make the calculations easier o When dividing make the numerator larger than the denominator through the manipulation of scientific notation o When doing addition and subtraction bring the numbers to the same power of magnitude. Example 1: 0.0005/0.0006= Example 2: 2.20 x 10-­‐18 -­‐2.20 x 10-­‐19 = Problem 1: 0.0056
8 Problem 2: 1.25 x 10-­‐9 -­‐ 1.25 x 10-­‐8 Problem 3: 0.00176
0.44 Problem 4: (0.2) × (0.004) × (0.56)
(0.3) × (0.16) × (0.14)
B. Manipulating Exponents •
(ab)n = anbn •
anam=an+m •
an
⎛ a ⎞
⎜ ⎟ = n b
⎝ b ⎠
•
a −n =
•
⎛ a ⎞
⎜ ⎟
⎝ b ⎠
n
Example 3: 1
an
−n
=
bn
a n 1
2 × 10 −2 Problem 5: 3 −3
6 −2
q
C. Calculating Square Roots ( in the format of (a
⇥ 10
b ) ) •
•
•
Make sure that number a and power of magnitude b can accommodate the next two steps. Make sure that your power of the magnitude (b) is divisible by 2 Decide what is the nearest complete square to your number ”a “ (Example 50 nearest number is 49=72) Example 4: 15 ×10 5 = Problem 7: (0.012)1/2 D. Calculating Cubic Roots • Make sure that your power of the magnitude is divisible by 3 • Determine the nearest complete cube to your number “a” Problem 8: (6x104)1/3 E. Simplify, Simplify, Simplify: • Find common denominators for fractions • Separate numbers from scientific notation Determining if a number is divisible by 2, 3, 5, 9, or 10: Example 5: Out of numbers below which are divisible by 2, 3, 5, 9, 10? All of them, some of them, or none of them? 75465 2775936 56002170 6588 123765 67845 233 Problem 9: : Find a number that contains only the number 2 that is divisible by 3. Problem 10: : Find a number that contains only the number 2 that is divisible by 9. F. General Guidelines: • Keep everything as fraction until the very end and then convert to decimal format • Convert percentages into fractions Example 6: You have several isotopes for an unknown compound 10.% is 5.00g, 25% is 4.00g, 35% is 3.00g, 30.% is 6.00g what is an atomic mass? (Answer: 4.65g) G. Rounding: • Common Constants: o Speed of light c=2.9979 ·∙108 m/s o Plank Constant h=6.626·∙10-­‐34 J·∙s o Avogadro number NA=6.02214 mol-­‐1 o Gas constant R=8.314 J/(mol·∙K)= 8.314 L·∙kPA/(mol·∙K) =0.08206 (L·∙atm)/(mol·∙K)= 0.08314(L·∙bar)/(mol·∙K) o Heat capacity of water c=4.184 J/(K·∙g) • Separate units from numbers Example 7: Using the formula, PV=nRT, and R constant above solve the following problem. What volume will 3.0 moles of an ideal gas occupy at 300.K and 2.0atm? (Answer:37 L) Example 8: Using the formula that ΔΕlight=hνlight= hc/λlight and necessary constants above calculate ΔΕlight of a light with wavelength of 330nm, 300nm, 600nm. (Answers: 6.0·∙10-­‐19J, 7·∙10-­‐19J, 3·∙10-­‐19J) Problem 11: (16.6x10-­‐7 -­‐ 1.2x10-­‐6)1/2 Problem 12:
0.88 × 0.6 × 9.0
0.011× 0.005 ×1.8 Problem 13: 186
279 Problem 14: 22 ×125 × 75 × 240 ×198
66 × 200 × 360 ×132 Problem 15: 150 × 75 × 140 × 330
90 × 120 × 210 × 495 Problem 16: 4 +
4
7
+3+
33
33 Problem 17: 5 +
13
7
− (2 + )
18
18 Problem 18: 5 ⋅ 24 + 7 ⋅16
5 ⋅ 24 − 7 ⋅16 Problem 19: (3.5)2 - (2.5)2
0.80 × 0.6 − 0.6 16b
4−b 4+b
4
Problem 20 : ( (16 − b 2 ) + 4 + b ) ÷ 4 − 4 − b
Problem 21: (1.25)-­‐9 x (1.25)8 x 0.2-­‐2 Answers to the problems: 7·10-4, 4.0·10-3, 6.7·10-2≈7·10-2, -11.25·10-9, 1.3≈1, 11·10-2, 40. ,
≈7·10-2, 48·103 ≈5·104, 0.666, 15.6≈16, 0.46, 7.3, 3.3, 29, -50, 0, 20