M
Math Coach Amy
Amy Ferguson Moncure

IB · Unit 1 · 1B Scientific Notation

Standard Form with a, b, c, and d

Substitute, then rewrite. c is the coefficient between 1 and 10. d is the matching power of 10, sometimes in terms of p and q.

Jump to section:

Key Ideas IB wording Calculator Easy Medium Hard

\(x^2 \cdot x^3 = x^5\)

\((x^2)^3 = x^6\)

3−2 = 19

What to Remember

Show that

Substitute the given values, multiply or divide, then rewrite until the coefficient is between 1 and 10. That should match the stated answer.

If a = 2 × 10³ and b = 4 × 10², show that a × b = 8 × 10⁵.

a×b

2 × 10³×4 × 10²

Rewrite using the commutative property of multiplication.

2 · 4 · 10³ · 10²

8 × 10⁵

Letters in the exponent

Treat p and q like numbers. Add them when you multiply. Subtract them when you divide.

(2 × 10p)(3 × 10q)

Rewrite using the commutative property of multiplication.

2 · 3 · 10p · 10q

6 × 10p+q

The 1 ≤ c < 10 rule

c is never 12 or 0.4. Pull out an extra factor of 10 so c lands in that interval.

12 × 10p+q

1.2 × 10¹ × 10p+q

1.2 × 10p+q+1

IB wording = scientific notation

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).

TI-84 Plus CE: turn on SCI
  1. Press MODE.
  2. On the first line (NORMAL SCI ENG), arrow right to SCI.
  3. Press ENTER so SCI is highlighted.
  4. Press 2nd then MODE (QUIT) to return to the home screen.

Answers will look like 3.2E4. That means \(3.2 \times 10^4\). Write the × 10 form on paper.

Level 1

Easy

Plug in the values and simplify. The coefficient should already work.

Example
If a = 2 × 10³ and b = 4 × 10², show that a × b = 8 × 10⁵.

a×b

2 × 10³×4 × 10²

Rewrite using the commutative property of multiplication.

2 · 4 · 10³ · 10²

8 × 10⁵

Answer: 8 × 10⁵

1
If a = 3 × 10⁴ and b = 2 × 10¹, show that a × b = 6 × 10⁵.
Show Answer

a×b

3 × 10⁴×2 × 10¹

Rewrite using the commutative property of multiplication.

3 · 2 · 10⁴ · 10¹

6 × 10⁵

Answer: 6 × 10⁵

2
If a = 9 × 10⁶ and b = 3 × 10², show that a ÷ b = 3 × 10⁴.
Show Answer

ab

9 × 10⁶3 × 10²

93 × 10⁶10²

Treat 93 as a fraction and reduce it to 3.

3 × 10⁴

Answer: 3 × 10⁴

3
If a = 5 × 10⁻¹ and b = 2 × 10⁴, show that a × b = 1 × 10⁴.
Show Answer

a×b

5 × 10⁻¹×2 × 10⁴

Rewrite using the commutative property of multiplication.

5 · 2 · 10⁻¹ · 10⁴

10 × 10³

1 × 10¹ × 10³

1 × 10⁴

Answer: 1 × 10⁴

4
If a = 8 × 10⁷ and b = 4 × 10⁷, show that a − b = 4 × 10⁷.
Show Answer

a−b

8 × 10⁷−4 × 10⁷

Same exponent, so subtract the coefficients.

4 × 10⁷

Answer: 4 × 10⁷

5
If a = 1.5 × 10³ and b = 2 × 10³, show that a + b = 3.5 × 10³.
Show Answer

a+b

1.5 × 10³+2 × 10³

Same exponent, so add the coefficients.

3.5 × 10³

Answer: 3.5 × 10³

Level 2

Medium

You will need one rewrite so the coefficient is between 1 and 10.

Example
If a = 4 × 10⁶ and b = 5 × 10³, show that a × b = 2 × 10¹⁰.

a×b

4 × 10⁶×5 × 10³

Rewrite using the commutative property of multiplication.

4 · 5 · 10⁶ · 10³

20 × 10⁹

2 × 10¹ × 10⁹

2 × 10¹⁰

Answer: 2 × 10¹⁰

1
If a = 7 × 10⁴ and b = 2 × 10⁵, show that a × b = 1.4 × 10¹⁰.
Show Answer

a×b

7 × 10⁴×2 × 10⁵

Rewrite using the commutative property of multiplication.

7 · 2 · 10⁴ · 10⁵

14 × 10⁹

1.4 × 10¹ × 10⁹

1.4 × 10¹⁰

Answer: 1.4 × 10¹⁰

2
If a = 3 × 10⁸ and b = 6 × 10², show that a ÷ b = 5 × 10⁵.
Show Answer

ab

3 × 10⁸6 × 10²

36 × 10⁸10²

Treat 36 as a fraction and reduce it to 0.5.

0.5 × 10⁶

5 × 10⁻¹ × 10⁶

5 × 10⁵

Answer: 5 × 10⁵

3
If a = 8.4 × 10⁵ and b = 2.1 × 10², show that a ÷ b = 4 × 10³.
Show Answer

ab

8.4 × 10⁵2.1 × 10²

8.42.1 × 10⁵10²

Treat 8.42.1 as a fraction and reduce it to 4.

4 × 10³

Answer: 4 × 10³

4
If a = 6.2 × 10⁷ and b = 4.8 × 10⁶, show that a − b = 5.72 × 10⁷.
Show Answer

a−b

6.2 × 10⁷−4.8 × 10⁶

Change the larger power to the smaller power.

6.2 × 10⁷−4.8 × 10⁶=62 × 10⁶−4.8 × 10⁶

57.2 × 10⁶

5.72 × 10¹ × 10⁶

5.72 × 10⁷

Answer: 5.72 × 10⁷

5
If a = 9 × 10⁻³ and b = 4 × 10⁻⁴, show that a + b = 9.4 × 10⁻³.
Show Answer

a+b

9 × 10⁻³+4 × 10⁻⁴

Change the larger power to the smaller power.

9 × 10⁻³+4 × 10⁻⁴=90 × 10⁻⁴+4 × 10⁻⁴

94 × 10⁻⁴

9.4 × 10¹ × 10⁻⁴

9.4 × 10⁻³

Answer: 9.4 × 10⁻³

Level 3

Hard

Leave d in terms of p and q. Watch whether the coefficient forces d to change by 1.

Example
You are given that (2 × 10p)(7 × 10q) = c × 10d where 1 ≤ c < 10 and k ⊂ ℤ.
a Find the value of c.
b Find an expression for d in terms of p and q.

a×b

2 × 10p×7 × 10q

Rewrite using the commutative property of multiplication.

2 · 7 · 10p · 10q

14 × 10p+q

1.4 × 10¹ × 10p+q

1.4 × 10p+q+1

a c = 1.4

b d = p + q + 1

Answer: a c = 1.4    b d = p + q + 1

1
You are given 8 × 10p2 × 10q = c × 10d where 1 ≤ c < 10. Find c. Find d in terms of p and q.
Show Answer

ab

8 × 10p2 × 10q

82 × 10p10q

Treat 82 as a fraction and reduce it to 4.

4 × 10p−q

c = 4, d = p − q.

Answer: c = 4, d = p − q

2
You are given that (8 × 10p)(2 × 10q) = c × 10d where 1 ≤ c < 10 and k ⊂ ℤ.
a Find the value of c.
b Find an expression for d in terms of p and q.
Show Answer

a×b

8 × 10p×2 × 10q

Rewrite using the commutative property of multiplication.

8 · 2 · 10p · 10q

16 × 10p+q

1.6 × 10¹ × 10p+q

1.6 × 10p+q+1

a c = 1.6

b d = p + q + 1

Answer: a c = 1.6    b d = p + q + 1

3
You are given 3 × 10p6 × 10q = c × 10d where 1 ≤ c < 10. Find c. Find d in terms of p and q.
Show Answer

ab

3 × 10p6 × 10q

36 × 10p10q

Treat 36 as a fraction and reduce it to 0.5.

0.5 × 10p−q

5 × 10⁻¹ × 10p−q

5 × 10p−q−1

c = 5, d = p − q − 1.

Answer: c = 5, d = p − q − 1

4
x = a × 10p and y = b × 10q where 5 ≤ a < 10 and 5 ≤ b < 10. When xy is written in standard form as c × 10r, find an expression for r in terms of p and q.
Show Answer

x×y

a × 10p×b × 10q

Rewrite using the commutative property of multiplication.

a · b · 10p · 10q

ab × 10p+q

ab is at least 25 and less than 100, so 10 ≤ ab < 100.

ab/10 × 10¹ × 10p+q

c × 10p+q+1

r = p + q + 1

Answer: r = p + q + 1

5
You are given (7 × 10p − 3 × 10p)2 × 10q = c × 10d where 1 ≤ c < 10. Find c. Find d in terms of p and q.
Show Answer

a−bc

7 × 10p−3 × 10p2 × 10q

Same exponent in the numerator, so subtract the coefficients.

4 × 10p2 × 10q

42 × 10p10q

Treat 42 as a fraction and reduce it to 2.

2 × 10p−q

c = 2, d = p − q.

Answer: c = 2, d = p − q

Need help with standard-form show-that questions?

Book a Session with Amy