IB · Unit 1 · 1B Scientific Notation
Match the powers of 10 first. Then add or subtract the coefficients, and put the answer back into standard form.
Jump to section:
\(x^2 \cdot x^3 = x^5\)
\((x^2)^3 = x^6\)
3−2 = 19
Same power of 10 first
You can only add and subtract (combine) like terms, so the powers need to be the same.
Strategy: change the larger power to the smaller power.
2 × 10³+5 × 10⁴=2 × 10³+50 × 10³
52 × 10³
5.2 × 10¹ × 10³
5.2 × 10⁴
Add or subtract the coefficients only
2 × 10³+50 × 10³=52 × 10³
Do not add the exponents. That would be a multiply, not an add.
Subtraction can be negative
If you subtract a number with a much larger exponent, the answer is negative. Rewrite to the larger power, then subtract.
You may check on the TI-84 in SCI mode.
When a question says write the answer in the form a × 10ᵏ where 1 ≤ a < 10 and k ⊂ ℤ, that is the IB way of saying write the answer in scientific notation.
k ⊂ ℤ means k is an integer (k is in the set of integers).
Answers will look like 3.2E4. That means \(3.2 \times 10^4\). Write the × 10 form on paper.
The powers of 10 already match.
Same exponent, so add the coefficients.
3 + 2 = 5.
5 × 10⁴
Answer: 5 × 10⁴
6 + 1 = 7.
7 × 10⁵
Answer: 7 × 10⁵
9 − 4 = 5.
5 × 10³
Answer: 5 × 10³
2.1 + 3.4 = 5.5.
5.5 × 10⁶
Answer: 5.5 × 10⁶
8 − 3 = 5.
5 × 10⁻²
Answer: 5 × 10⁻²
4.7 + 2.2 = 6.9.
6.9 × 10⁸
Answer: 6.9 × 10⁸
The exponents are different. Change the larger power to the smaller power, then add or subtract the coefficients.
Change the larger power to the smaller power.
2 × 10³+5 × 10⁴=2 × 10³+50 × 10³
52 × 10³
5.2 × 10¹ × 10³
5.2 × 10⁴
Answer: 5.2 × 10⁴
Change the larger power to the smaller power.
4 × 10³+6 × 10⁴=4 × 10³+60 × 10³
64 × 10³
6.4 × 10¹ × 10³
6.4 × 10⁴
Answer: 6.4 × 10⁴
Change the larger power to the smaller power.
5 × 10⁶+3 × 10⁷=5 × 10⁶+30 × 10⁶
35 × 10⁶
3.5 × 10¹ × 10⁶
3.5 × 10⁷
Answer: 3.5 × 10⁷
Change the larger power to the smaller power.
2 × 10⁹+8 × 10⁷=200 × 10⁷+8 × 10⁷
208 × 10⁷
2.08 × 10² × 10⁷
2.08 × 10⁹
Answer: 2.08 × 10⁹
Change the larger power to the smaller power.
7 × 10²+1 × 10⁴=7 × 10²+100 × 10²
107 × 10²
1.07 × 10² × 10²
1.07 × 10⁴
Answer: 1.07 × 10⁴
Change the larger power to the smaller power.
3 × 10⁵+9 × 10⁶=3 × 10⁵+90 × 10⁵
93 × 10⁵
9.3 × 10¹ × 10⁵
9.3 × 10⁶
Answer: 9.3 × 10⁶
Change the larger power to the smaller power.
7 × 10⁶−2 × 10⁵=70 × 10⁵−2 × 10⁵
68 × 10⁵
6.8 × 10¹ × 10⁵
6.8 × 10⁶
Answer: 6.8 × 10⁶
Change the larger power to the smaller power.
5 × 10⁴−8 × 10³=50 × 10³−8 × 10³
42 × 10³
4.2 × 10¹ × 10³
4.2 × 10⁴
Answer: 4.2 × 10⁴
Change the larger power to the smaller power.
6 × 10⁸−9 × 10⁶=600 × 10⁶−9 × 10⁶
591 × 10⁶
5.91 × 10² × 10⁶
5.91 × 10⁸
Answer: 5.91 × 10⁸
Change the larger power to the smaller power.
4.5 × 10⁻³+6 × 10⁻⁴=45 × 10⁻⁴+6 × 10⁻⁴
51 × 10⁻⁴
5.1 × 10¹ × 10⁻⁴
5.1 × 10⁻³
Answer: 5.1 × 10⁻³
Change the larger power to the smaller power.
1.2 × 10⁸−5 × 10⁶=120 × 10⁶−5 × 10⁶
115 × 10⁶
1.15 × 10² × 10⁶
1.15 × 10⁸
Answer: 1.15 × 10⁸
Change the larger power to the smaller power. If you subtract a much bigger term, the answer is negative. Then fix the coefficient if it is 10 or more.
Change the larger power to the smaller power.
6 × 10⁶−5 × 10⁵=60 × 10⁵−5 × 10⁵
55 × 10⁵
5.5 × 10¹ × 10⁵
5.5 × 10⁶
Answer: 5.5 × 10⁶
Change the larger power to the smaller power.
3.2 × 10⁶−8 × 10⁵=32 × 10⁵−8 × 10⁵
24 × 10⁵
2.4 × 10¹ × 10⁵
2.4 × 10⁶
Answer: 2.4 × 10⁶
Change the larger power to the smaller power.
9.1 × 10⁷−9.1 × 10⁶=91 × 10⁶−9.1 × 10⁶
81.9 × 10⁶
8.19 × 10¹ × 10⁶
8.19 × 10⁷
Answer: 8.19 × 10⁷
The second term is much larger, so the answer will be negative.
Change the larger power to the smaller power.
3 × 10²−5 × 10⁸=3 × 10²−5 000 000 × 10²
-4 999 997 × 10²
-4.999997 × 10⁶ × 10²
-4.999997 × 10⁸
Answer: -4.999997 × 10⁸
Same exponent, so add the coefficients.
14.5 × 10³
1.45 × 10¹ × 10³
1.45 × 10⁴
Answer: 1.45 × 10⁴
2.6 + 9.5 = 12.1, so 12.1 × 10⁵.
Change the larger power to the smaller power.
12.1 × 10⁵−4 × 10⁴=121 × 10⁴−4 × 10⁴
117 × 10⁴
1.17 × 10² × 10⁴
1.17 × 10⁶
Answer: 1.17 × 10⁶
The second term is much larger, so the answer will be negative.
Change the larger power to the smaller power.
4 × 10³−7 × 10⁷=4 × 10³−70 000 × 10³
-69 996 × 10³
-6.9996 × 10⁴ × 10³
-6.9996 × 10⁷
Answer: -6.9996 × 10⁷
Change the larger power to the smaller power.
5 × 10⁻⁴+2 × 10⁻⁴+8 × 10⁻⁵=50 × 10⁻⁵+20 × 10⁻⁵+8 × 10⁻⁵
78 × 10⁻⁵
7.8 × 10¹ × 10⁻⁵
7.8 × 10⁻⁴
Answer: 7.8 × 10⁻⁴
Need help adding and subtracting in standard form?
Book a Session with Amy