Lesson 17 ~25 min Unit 1 · Index Laws +85 XP

Operations in Scientific Notation

Multiply, divide, add and subtract numbers in $a \times 10^n$ form — using index laws on the powers and arithmetic on the coefficients.

Today's hook: $(3 \times 10^4) \times (2 \times 10^5)$ — do you need a calculator? No. Multiply the coefficients, add the powers, and you're done.

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Think First

warm-up

Predict $(3 \times 10^4) \times (2 \times 10^5)$. What index law tells you what to do with the $10^4$ and $10^5$? What do you do with the $3$ and the $2$?

Record in your workbook.

The Big Idea

+5 XP

For $\times$ and $\div$ in scientific notation, deal with the coefficients using arithmetic and the powers of $10$ using index laws. For $+$ and $-$, the powers of $10$ must match first.

Multiply: $\times$ coefficients, add indices. Divide: $\div$ coefficients, subtract indices. Add/subtract: line up the powers first — only then add or subtract the coefficients.

$(3 \times 10^4)(2 \times 10^5) = 6 \times 10^9$

$\times$ — add indices

$10^m \cdot 10^n = 10^{m+n}$

$\div$ — subtract indices

$\dfrac{10^m}{10^n} = 10^{m-n}$

$+$, $-$ — same power first

Then add/subtract coefficients.

What You'll Master

objectives

Know

$\times$: multiply coefficients, add indices
$\div$: divide coefficients, subtract indices
$+/-$: rewrite to a common power, then operate

Understand

How index laws underpin every scientific-notation operation
Why you re-standardise if the coefficient leaves $[1, 10)$
Why addition needs matching powers

Can Do

$(3 \times 10^4)(2 \times 10^5) = 6 \times 10^9$
$\dfrac{8 \times 10^6}{2 \times 10^2} = 4 \times 10^4$
$3 \times 10^4 + 2 \times 10^3 = 3.2 \times 10^4$

Words You Need

vocabulary

Coefficient arithmeticNormal $\times$/$\div$ on the $a$ part.

Index law on powersAdd for $\times$, subtract for $\div$.

Re-standardiseShift the coefficient back into $[1, 10)$ and adjust the power.

Common powerA single $10^n$ both terms share before $+$ or $-$.

Like termsSame power of $10$ acts like the same pronumeral — coefficients combine.

Order of operationsPowers and brackets first, then $\times$/$\div$, then $+$/$-$.

Spot the Trap

heads-up

Wrong: “$(3 \times 10^4)(2 \times 10^5) = 6 \times 10^{20}$” — multiplying the indices instead of adding.

Right: $6 \times 10^{4+5} = 6 \times 10^9$. Multiplication of bases adds the indices.

Wrong: “$3 \times 10^4 + 2 \times 10^3 = 5 \times 10^7$” — adding both the coefficients and the indices.

Right: Rewrite to a common power: $3 \times 10^4 + 0.2 \times 10^4 = 3.2 \times 10^4$.

Multiply and divide

+5 XP

Use $(a \times 10^m)(b \times 10^n) = (a b) \times 10^{m+n}$ and $\dfrac{a \times 10^m}{b \times 10^n} = \dfrac{a}{b} \times 10^{m-n}$. If the new coefficient drifts out of $[1, 10)$, re-standardise.

$(5 \times 10^3)(4 \times 10^2) = 20 \times 10^5 = 2 \times 10^6$ — the $20$ became $2$ by shifting one place and bumping the power. $\dfrac{8 \times 10^6}{2 \times 10^2} = 4 \times 10^4$.

$\dfrac{8 \times 10^6}{2 \times 10^2} = 4 \times 10^4$

Add and subtract

+5 XP

The powers of $10$ must match first. Rewrite the smaller-power term to share the larger power, then add or subtract coefficients.

$3 \times 10^4 + 2 \times 10^3$. Rewrite $2 \times 10^3 = 0.2 \times 10^4$. Now: $(3 + 0.2) \times 10^4 = 3.2 \times 10^4$. For subtraction: $5 \times 10^{-3} - 2 \times 10^{-4} = 5 \times 10^{-3} - 0.2 \times 10^{-3} = 4.8 \times 10^{-3}$.

$3 \times 10^4 + 2 \times 10^3 = 3.2 \times 10^4$

Watch Me Solve It · 3 examples

Watch Me Solve It · Multiply

+15 XP per step

PROBLEM

Calculate $(5 \times 10^3) \times (4 \times 10^2)$, giving your answer in scientific notation.

1
Multiply the coefficients
$5 \times 4 = 20$
2
Add the indices
$10^{3+2} = 10^5$ → current: $20 \times 10^5$
3
Re-standardise the coefficient
$20 \times 10^5 = 2 \times 10^6$ (shift one left, bump power up)

Answer$2 \times 10^6$

Watch Me Solve It · Divide

+15 XP per step

PROBLEM

Calculate $\dfrac{8 \times 10^6}{2 \times 10^2}$.

1
Divide the coefficients
$\dfrac{8}{2} = 4$
2
Subtract the indices
$10^{6-2} = 10^4$
3
Combine
$4 \times 10^4$ (coefficient already in $[1, 10)$)

Answer$4 \times 10^4$

Watch Me Solve It · Add

+15 XP per step

PROBLEM

Calculate $3 \times 10^4 + 2 \times 10^3$, giving the answer in scientific notation.

1
Rewrite to a common power
$2 \times 10^3 = 0.2 \times 10^4$
2
Add the coefficients
$3 + 0.2 = 3.2$
3
Write the result
$3.2 \times 10^4$ (already standardised)

Answer$3.2 \times 10^4$

Common Pitfalls

heads-up

Multiplying the powers instead of adding

$(3 \times 10^4)(2 \times 10^5) = 6 \times 10^9$, not $6 \times 10^{20}$.

Fix: $\times$ of like bases adds indices.

Adding indices when adding terms

$3 \times 10^4 + 2 \times 10^3 \ne 5 \times 10^7$.

Fix: Match the powers first, then add the coefficients.

Leaving an unstandardised answer

$20 \times 10^5$ is the right value but the wrong form.

Fix: Shift the coefficient back into $[1, 10)$ and adjust the power.

Copy Into Your Books

Multiply

$(a \times 10^m)(b \times 10^n) = a b \times 10^{m+n}$
$(5 \times 10^3)(4 \times 10^2) = 2 \times 10^6$

Divide

$\dfrac{a \times 10^m}{b \times 10^n} = \dfrac{a}{b} \times 10^{m-n}$
$\dfrac{8 \times 10^6}{2 \times 10^2} = 4 \times 10^4$

Add/subtract

Make powers match
$3 \times 10^4 + 2 \times 10^3 = 3.2 \times 10^4$

Re-standardise

Keep $a$ in $[1, 10)$
$20 \times 10^5 \to 2 \times 10^6$

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Operations drills

4 problems

Combine coefficients with arithmetic and powers with index laws.

1 $(3 \times 10^4)(2 \times 10^5)$.
$3 \cdot 2 = 6$; $10^{4+5} = 10^9$.$6 \times 10^9$
2 $\dfrac{9 \times 10^8}{3 \times 10^3}$.
$9/3 = 3$; $10^{8-3} = 10^5$.$3 \times 10^5$
3 $5 \times 10^{-3} - 2 \times 10^{-4}$.
Rewrite $2 \times 10^{-4} = 0.2 \times 10^{-3}$; subtract: $(5 - 0.2) \times 10^{-3} = 4.8 \times 10^{-3}$.$4.8 \times 10^{-3}$
4 $(6 \times 10^5)(5 \times 10^{-2})$.
$6 \cdot 5 = 30$; $10^{5 + (-2)} = 10^3$; re-standardise $30 \times 10^3 = 3 \times 10^4$.$3 \times 10^4$

Complete in your workbook.

Quick Check · 5 questions

$(3 \times 10^4)(2 \times 10^5) = $

+10 XP

$\dfrac{8 \times 10^6}{2 \times 10^2} = $

+10 XP

$3 \times 10^4 + 2 \times 10^3 = $

+10 XP

$(6 \times 10^5)(5 \times 10^{-2}) = $

+10 XP

$5 \times 10^{-3} - 2 \times 10^{-4} = $

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyEasy3 MARKS

Q6. Calculate, leaving each in scientific notation: (a) $(4 \times 10^3)(2 \times 10^5)$, (b) $\dfrac{6 \times 10^9}{3 \times 10^4}$, (c) $(5 \times 10^{-2})(3 \times 10^6)$.

Answer in your workbook.

ApplyMedium3 MARKS

Q7. Calculate, giving each in scientific notation: (a) $4 \times 10^5 + 3 \times 10^4$, (b) $6 \times 10^{-2} - 5 \times 10^{-3}$, (c) $(8 \times 10^4)(5 \times 10^3)$ (re-standardise).

Answer in your workbook.

ReasonHard3 MARKS

Q8. Light travels at about $3 \times 10^8$ m/s. The Sun is about $1.5 \times 10^{11}$ m from Earth. (a) How long, in seconds, does sunlight take to reach Earth? Give your answer in scientific notation. (b) Express that time in minutes (to one decimal place).

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $6 \times 10^9$.

2. A — $4 \times 10^4$.

3. C — $3.2 \times 10^4$.

4. D — $3 \times 10^4$.

5. A — $4.8 \times 10^{-3}$.

Show Your Working Model Answers

Q6 (3 marks): (a) $4 \cdot 2 = 8$; $10^{3+5} = 10^8$; $8 \times 10^8$ [1]. (b) $6/3 = 2$; $10^{9-4} = 10^5$; $2 \times 10^5$ [1]. (c) $5 \cdot 3 = 15$; $10^{-2 + 6} = 10^4$; $15 \times 10^4 = 1.5 \times 10^5$ [1].

Q7 (3 marks): (a) Rewrite $3 \times 10^4 = 0.3 \times 10^5$; sum $4.3 \times 10^5$ [1]. (b) Rewrite $5 \times 10^{-3} = 0.5 \times 10^{-2}$; $6 - 0.5 = 5.5$; $5.5 \times 10^{-2}$ [1]. (c) $8 \cdot 5 = 40$; $10^{4+3} = 10^7$; $40 \times 10^7 = 4 \times 10^8$ [1].

Q8 (3 marks): (a) $t = \dfrac{1.5 \times 10^{11}}{3 \times 10^8} = 0.5 \times 10^3 = 5 \times 10^2$ s [2]. (b) $5 \times 10^2 \div 60 = 500/60 \approx 8.3$ minutes [1].

Stretch Challenge · +25 XP, +10 coins

Multi-Step Mission

Earth has about $7.8 \times 10^9$ people. Each person breathes roughly $6 \times 10^3$ litres of air per day. (a) How many litres are breathed in by humanity in one day? Give your answer in scientific notation. (b) In one year ($365$ days)? (c) Name each index law you used.

Reveal solution

(a) $(7.8 \times 10^9)(6 \times 10^3) = 46.8 \times 10^{12} = 4.68 \times 10^{13}$ L per day. (b) $\times 365$ days: $4.68 \times 10^{13} \cdot 3.65 \times 10^2 = 17.08 \times 10^{15} \approx 1.71 \times 10^{16}$ L/year. Laws used: product rule (multiply coefficients, add indices) and re-standardisation (shift coefficient back to $[1, 10)$ and bump the power).

Quick Review

$\times$

Coeffs $\times$, indices $+$

$\div$

Coeffs $\div$, indices $-$

$+$, $-$

Match powers first

Re-standardise

Keep $1 \leq a < 10$

Example $\times$

$(3 \times 10^4)(2 \times 10^5) = 6 \times 10^9$

Example $+$

$3 \times 10^4 + 2 \times 10^3 = 3.2 \times 10^4$

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