Multiply and Divide in Standard Form

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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

Multiply

Multiply the coefficients. Add the exponents. Then rewrite so the coefficient is between 1 and 10.

(a × 10^m)(b × 10ⁿ) = (ab) × 10^m+n

(2 × 10³)(4 × 10⁵)

Rewrite using the commutative property of multiplication.

2 · 4 · 10³ · 10⁵

8 × 10⁸

Divide

Divide the coefficients. Subtract the exponents. Then rewrite if needed.

8 × 10⁷2 × 10²

82 × 10⁷10²

Treat 82 as a fraction and reduce it to 4.

4 × 10⁵

Fix the coefficient

If ab is 10 or more, change that part to scientific notation.

(4 × 10³)(5 × 10⁴)

Rewrite using the commutative property of multiplication.

4 · 5 · 10³ · 10⁴

20 × 10⁷

2 × 10¹ × 10⁷

2 × 10⁸

Finding c and d

IB often asks you to write the answer as c × 10^d with 1 ≤ c < 10. c is the cleaned-up coefficient. d may be an expression in p and q.

(2 × 10^p)(3 × 10^q) = c × 10^d

Rewrite using the commutative property of multiplication.

2 · 3 · 10^p · 10^q

6 × 10^p+q

c = 6

d = p + q

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

Press MODE.
On the first line (NORMAL SCI ENG), arrow right to SCI.
Press ENTER so SCI is highlighted.
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

The coefficient stays between 1 and 10. No extra rewrite.

Example

Simplify. (2 × 10³)(4 × 10⁵)

Rewrite using the commutative property of multiplication.

2 · 4 · 10³ · 10⁵

8 × 10⁸

Answer: 8 × 10⁸

Simplify. (3 × 10²)(2 × 10⁴)

Show Answer

Rewrite using the commutative property of multiplication.

3 · 2 · 10² · 10⁴

6 × 10⁶

Answer: 6 × 10⁶

Simplify. (5 × 10⁶)(1 × 10³)

Show Answer

Rewrite using the commutative property of multiplication.

5 · 1 · 10⁶ · 10³

5 × 10⁹

Answer: 5 × 10⁹

Simplify. 8 × 10⁷2 × 10²

Show Answer

82 × 10⁷10²

Treat 82 as a fraction and reduce it to 4.

4 × 10⁵

Answer: 4 × 10⁵

Simplify. 9 × 10⁸3 × 10⁴

Show Answer

93 × 10⁸10⁴

Treat 93 as a fraction and reduce it to 3.

3 × 10⁴

Answer: 3 × 10⁴

Simplify. (4 × 10⁻²)(2 × 10⁵)

Show Answer

Rewrite using the commutative property of multiplication.

4 · 2 · 10⁻² · 10⁵

8 × 10³

Answer: 8 × 10³

Level 2

Medium

The coefficient lands outside 1 to 10. Rewrite it.

Example

Simplify. (4 × 10³)(5 × 10⁴)

Rewrite using the commutative property of multiplication.

4 · 5 · 10³ · 10⁴

20 × 10⁷

2.0 × 10¹ × 10⁷

2.0 × 10⁸

Answer: 2.0 × 10⁸

Simplify. (6 × 10⁵)(4 × 10²)

Show Answer

Rewrite using the commutative property of multiplication.

6 · 4 · 10⁵ · 10²

24 × 10⁷

2.4 × 10¹ × 10⁷

2.4 × 10⁸

Answer: 2.4 × 10⁸

Simplify. (3.2 × 10⁴)(5 × 10³)

Show Answer

Rewrite using the commutative property of multiplication.

3.2 · 5 · 10⁴ · 10³

16 × 10⁷

1.6 × 10¹ × 10⁷

1.6 × 10⁸

Answer: 1.6 × 10⁸

Simplify. 1.5 × 10⁶5 × 10²

Show Answer

1.55 × 10⁶10²

Treat 1.55 as a fraction and reduce it to 0.3.

0.3 × 10⁴

3 × 10⁻¹ × 10⁴

3 × 10³

Answer: 3 × 10³

Simplify. 8.4 × 10⁻³2 × 10⁻⁵

Show Answer

8.42 × 10⁻³10⁻⁵

Treat 8.42 as a fraction and reduce it to 4.2.

4.2 × 10²

Answer: 4.2 × 10²

Simplify. (7 × 10⁻⁴)(2 × 10⁻³)

Show Answer

Rewrite using the commutative property of multiplication.

7 · 2 · 10⁻⁴ · 10⁻³

14 × 10⁻⁷

1.4 × 10¹ × 10⁻⁷

1.4 × 10⁻⁶

Answer: 1.4 × 10⁻⁶

Level 3

Hard

Several factors, or write the answer as c × 10^d with 1 ≤ c < 10.

Example

Simplify. (4.5 × 10⁸)(6 × 10⁻³)

Rewrite using the commutative property of multiplication.

4.5 · 6 · 10⁸ · 10⁻³

27 × 10⁵

2.7 × 10¹ × 10⁵

2.7 × 10⁶

Answer: 2.7 × 10⁶

Simplify. (2.4 × 10⁻⁵)(5 × 10⁻²)

Show Answer

Rewrite using the commutative property of multiplication.

2.4 · 5 · 10⁻⁵ · 10⁻²

12 × 10⁻⁷

1.2 × 10¹ × 10⁻⁷

1.2 × 10⁻⁶

Answer: 1.2 × 10⁻⁶

Simplify. 9.6 × 10⁷3.2 × 10⁻⁴

Show Answer

9.63.2 × 10⁷10⁻⁴

Treat 9.63.2 as a fraction and reduce it to 3.

3 × 10¹¹

Answer: 3 × 10¹¹

Simplify. (5.5 × 10⁴)(8 × 10⁵)2 × 10³

Show Answer

5.5 · 82 × 10⁴ · 10⁵10³

Treat 442 as a fraction and reduce it to 22.

22 × 10⁶

2.2 × 10¹ × 10⁶

2.2 × 10⁷

Answer: 2.2 × 10⁷

You are given that (4 × 10^p)(3 × 10^q) = c × 10^d 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

Rewrite using the commutative property of multiplication.

4 · 3 · 10^p · 10^q

12 × 10^p+q

1.2 × 10¹ × 10^p+q

1.2 × 10^p+q+1

a c = 1.2

b d = p + q + 1

Answer: a c = 1.2 b d = p + q + 1

You are given that (4 × 10^p)(6 × 10^q) = c × 10^d 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

Rewrite using the commutative property of multiplication.

4 · 6 · 10^p · 10^q

24 × 10^p+q

2.4 × 10¹ × 10^p+q

2.4 × 10^p+q+1

a c = 2.4

b d = p + q + 1

Answer: a c = 2.4 b d = p + q + 1

You are given that (2 × 10^p)(3 × 10^q) = c × 10^d 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

Rewrite using the commutative property of multiplication.

2 · 3 · 10^p · 10^q

6 × 10^p+q

6 is already between 1 and 10, so do not rewrite.

a c = 6

b d = p + q

Answer: a c = 6 b d = p + q

You are given 2 × 10^p5 × 10^q = c × 10^d where 1 ≤ c < 10. Find c. Find an expression for d in terms of p and q.

Show Answer

25 × 10^p10^q

Treat 25 as a fraction and reduce it to 0.4.

0.4 × 10^p−q

4 × 10⁻¹ × 10^p−q

4 × 10^p−q−1

c = 4 and d = p − q − 1.

Answer: c = 4, d = p − q − 1

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