Lesson 20 ~30 min Unit 1 · Synthesis +95 XP

Unit Synthesis — Index Laws Review

All six index laws on one page, plus scientific notation and fractional indices. Mixed exam-style problems that combine everything you've learned in Unit 1.

Today's hook: Simplify $\dfrac{(2a^3)^2 \cdot a^{-1}}{4 a^{1/2}}$. You will need five different index laws to crack this single expression — product, quotient, power-of-a-product, negative, and fractional. Today is the day they all click.

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Think First

warm-up

List every index law you can remember from L01–L19. Try to write each one as a formula. Don't peek — you'll check your list against the next card.

Record in your workbook.

The Big Six — All laws on one page

+5 XP

Here are all six index laws covered in Unit 1. Memorise these cold — every question in this unit reduces to combining them.

1. Product: $x^m \cdot x^n = x^{m+n}$. 2. Quotient: $\dfrac{x^m}{x^n} = x^{m-n}$. 3. Power of a power: $(x^m)^n = x^{mn}$. 4. Power of a product: $(xy)^n = x^n y^n$. 5. Zero index: $x^0 = 1$ ($x \ne 0$). 6. Negative index: $x^{-n} = \dfrac{1}{x^n}$. Bonus: $x^{m/n} = \sqrt[n]{x^m}$.

$x^m \cdot x^n = x^{m+n}$, $\;\;(x^m)^n = x^{mn}$, $\;\;x^{-n} = \dfrac{1}{x^n}$

Order: brackets first

Powers $\to$ multiply $\to$ divide.

Negative $\to$ flip

$x^{-n}$ moves to the denominator.

Fraction $\to$ root

Bottom = root, top = power.

What You'll Master

objectives

Know

All six index laws (and the fractional bonus law)
Standard form $a \times 10^n$ with $1 \le a < 10$
Convention: most answers to 3 sig fig

Understand

How to chain laws together in one expression
When “root first” saves you arithmetic
Why scientific notation is a special case of the same laws

Can Do

Simplify mixed integer-and-fractional-index expressions
Calculate with scientific notation accurately
Solve word problems involving cosmic/atomic scales

Words You Need (revision)

vocabulary

BaseThe number being multiplied: in $x^n$, $x$ is the base.

Index / exponentThe power: in $x^n$, $n$ is the index.

CoefficientNumber in front of a variable: in $5 x^3$ it's $5$.

Scientific notation$a \times 10^n$ with $1 \le a < 10$.

Significant figuresThe digits carrying information; 3 s.f. is standard.

SurdIrrational root; e.g. $\sqrt{2} = 2^{1/2}$.

Top Traps from the Whole Unit

heads-up

Wrong: $(2 a)^3 = 2 a^3$ — forgetting the coefficient takes the power.

Right: $(2 a)^3 = 2^3 a^3 = 8 a^3$.

Wrong: $x^{-2} = -x^2$ — treating the negative as a sign on the base.

Right: $x^{-2} = \dfrac{1}{x^2}$. Negative index flips the base.

Wrong: Leaving a scientific-notation coefficient outside $[1, 10)$, e.g. $12 \times 10^9$.

Right: Re-normalise: $12 \times 10^9 = 1.2 \times 10^{10}$.

Lesson map — what we covered

+5 XP

Use this map to revise. Each row links a lesson to its key formula.

L01–L05: base, exponent, product/quotient/power laws — numbers. L06–L09: zero index, negative index, and converting between fractional and negative forms. L10–L13: applying laws to algebraic expressions, multi-variable terms. L14–L15: scientific notation for large/small numbers, converting to/from standard form. L16–L17: arithmetic in scientific notation, comparing orders of magnitude. L18: applications, sig fig, calculator. L19: fractional indices and surds.

$x^m \cdot x^n = x^{m+n}\;\;\dfrac{x^m}{x^n} = x^{m-n}\;\;(x^m)^n = x^{mn}\;\;x^0 = 1\;\;x^{-n} = \dfrac{1}{x^n}\;\;x^{m/n} = \sqrt[n]{x^m}$

A four-law expression done slowly

+5 XP

Simplify $\dfrac{(3 x^2)^2 \cdot x^{-3}}{9 x^0}$. We will need: power-of-a-product, product, negative index, zero index, and quotient.

Power first: $(3 x^2)^2 = 9 x^4$. Zero: $x^0 = 1$. Negative: $x^{-3} = \dfrac{1}{x^3}$, but keep as $x^{-3}$ for the product step. Product on numerator: $9 x^4 \cdot x^{-3} = 9 x^{4-3} = 9 x^1$. Quotient: $\dfrac{9 x}{9 \cdot 1} = x$.

$\dfrac{(3 x^2)^2 \cdot x^{-3}}{9 x^0} = x$

Watch Me Solve It · 3 mixed examples

Watch Me Solve It · Mixed integer & negative indices

+15 XP per step

PROBLEM

Simplify $\dfrac{6 a^5 b^{-2}}{2 a^2 b^{-5}}$, giving the answer with positive indices only.

1
Divide coefficients
$6 / 2 = 3$
2
Quotient rule on each variable
$a^{5-2} = a^3$; $\;b^{-2 - (-5)} = b^{3}$
3
Combine
$3 a^3 b^3$

Answer$3 a^3 b^3$

Watch Me Solve It · Scientific notation problem

+15 XP per step

PROBLEM

A computer performs $4.0 \times 10^9$ operations per second. How many operations in $2.5 \times 10^3$ seconds, to 2 sig fig?

1
Set up the product
$N = (4.0 \times 10^9) \times (2.5 \times 10^3)$
2
Multiply coefficients, add indices
$4.0 \times 2.5 = 10$; $\;10^{9+3} = 10^{12}$
3
Re-normalise to standard form (2 s.f.)
$10 \times 10^{12} = 1.0 \times 10^{13}$

Answer$1.0 \times 10^{13}$ operations

Watch Me Solve It · Full-house: integer, negative, fractional

+15 XP per step

PROBLEM

Simplify $\dfrac{(2 a^3)^2 \cdot a^{-1}}{4 a^{1/2}}$, with a positive index in the final answer.

1
Power of a product on the bracket
$(2 a^3)^2 = 4 a^6$
2
Product rule on the numerator
$4 a^6 \cdot a^{-1} = 4 a^{6 - 1} = 4 a^5$
3
Quotient rule with fractional index
$\dfrac{4 a^5}{4 a^{1/2}} = a^{5 - 1/2} = a^{10/2 - 1/2} = a^{9/2}$

Answer$a^{9/2}$ (or $\sqrt{a^9}$)

Top Mistakes To Avoid In The Exam

heads-up

Multiplying indices when you should be adding

$x^2 \cdot x^3 \ne x^6$. Product rule adds indices.

Fix: Multiplying bases means adding indices; only the power-of-a-power law multiplies.

Misreading $x^0$

$x^0 = 1$, not $0$. Don't drop the whole term.

Fix: $5 x^0 = 5$, not $5 \cdot 0 = 0$.

Final answer still has a negative index

If a question says “positive index only”, you must flip negatives to the denominator.

Fix: Last line check — re-read the question's instruction.

Scientific notation not in standard form

$12.5 \times 10^4$ is not standard.

Fix: $1.25 \times 10^5$ — check $1 \le a < 10$.

Copy Into Your Books — Master Formula Sheet

Integer indices

Product: $x^m \cdot x^n = x^{m+n}$
Quotient: $x^m / x^n = x^{m-n}$
Power: $(x^m)^n = x^{mn}$
Of a product: $(xy)^n = x^n y^n$

Zero & negative

$x^0 = 1$ ($x \ne 0$)
$x^{-n} = \dfrac{1}{x^n}$
$\left(\dfrac{x}{y}\right)^{-n} = \left(\dfrac{y}{x}\right)^n$

Fractional & scientific

$x^{1/n} = \sqrt[n]{x}$
$x^{m/n} = \sqrt[n]{x^m}$
$a \times 10^n$, $1 \le a < 10$

Order of operations

Brackets / powers $\to$ multiply $\to$ divide
Re-normalise scientific notation at the end
Final check: positive index? right sig fig?

How are you completing this lesson?

Brain Trainer · 5 mixed drills

Brain Trainer · Mixed unit drills

5 problems

One question from each strand: integer indices, zero/negative, algebraic, scientific notation, fractional.

1 Simplify $x^7 \cdot x^{-3}$ as a positive index.
$x^{7 + (-3)} = x^4$.$x^4$
2 Evaluate $5^0 + 5^{-1}$.
$5^0 = 1$; $5^{-1} = 1/5 = 0.2$; sum $= 1.2$.$1.2$ or $\dfrac{6}{5}$
3 Simplify $\dfrac{(3 x^2 y)^2}{9 x y}$.
Numerator $= 9 x^4 y^2$; quotient $= x^{4-1} y^{2-1} = x^3 y$.$x^3 y$
4 Calculate $(2.5 \times 10^4) \times (4 \times 10^{-7})$.
$2.5 \times 4 = 10$; $10^{4-7} = 10^{-3}$; re-normalise $10 \times 10^{-3} = 1 \times 10^{-2}$.$1 \times 10^{-2}$ or $0.01$
5 Evaluate $16^{3/4}$.
$\sqrt[4]{16} = 2$; $2^3 = 8$.$8$

Complete in your workbook.

Quick Check · 5 questions (mixed)

Simplify $\dfrac{x^8}{x^3}$.

+10 XP

Evaluate $7^0$.

+10 XP

Expand $(3 x^2)^3$.

+10 XP

Calculate $(6 \times 10^5) \times (4 \times 10^{-2})$.

+10 XP

Evaluate $125^{2/3}$.

+10 XP

Show Your Working · 3 exam-style questions

Show Your Working

10 marks total

ApplyMedium3 MARKS

Q6. Simplify, expressing each answer with positive indices only: (a) $a^5 \cdot a^{-2}$, (b) $(2 x^3 y)^4$, (c) $\dfrac{12 m^2 n^{-1}}{4 m^{-3} n^2}$.

Answer in your workbook.

ApplyMedium3 MARKS

Q7. A red blood cell has a volume of $9.0 \times 10^{-14}$ L. An adult has about $5.0$ L of blood, of which $45\%$ is red blood cells. Estimate the number of red blood cells in their body, to 2 sig fig.

Answer in your workbook.

ReasonHard4 MARKS

Q8. (a) Simplify $\dfrac{(2 x^2)^3 \cdot x^{-1}}{4 x^{1/2}}$, stating each index law used. (b) Hence evaluate when $x = 4$.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. D — $x^5$.

2. B — $1$.

3. A — $27 x^6$.

4. C — $2.4 \times 10^4$.

5. D — $25$.

Show Your Working Model Answers

Q6 (3 marks): (a) $a^{5 + (-2)} = a^3$ [1]; (b) $(2 x^3 y)^4 = 2^4 \cdot x^{12} \cdot y^4 = 16 x^{12} y^4$ [1]; (c) $\dfrac{12}{4} = 3$; $m^{2 - (-3)} = m^5$; $n^{-1 - 2} = n^{-3}$ — answer with positive indices: $\dfrac{3 m^5}{n^3}$ [1].

Q7 (3 marks): Volume of RBCs = $5.0 \times 0.45 = 2.25$ L [1]. Number $= \dfrac{2.25}{9.0 \times 10^{-14}} = \dfrac{2.25}{9.0} \times 10^{14} = 0.25 \times 10^{14}$ [1] $= 2.5 \times 10^{13}$ red blood cells (2 s.f.) [1].

Q8 (4 marks): (a) Power of a product: $(2 x^2)^3 = 8 x^6$ [1]. Product on numerator: $8 x^6 \cdot x^{-1} = 8 x^5$ [1]. Quotient with fractional index: $\dfrac{8 x^5}{4 x^{1/2}} = 2 x^{5 - 1/2} = 2 x^{9/2}$ [1]. (b) At $x = 4$: $2 \cdot 4^{9/2} = 2 \cdot (\sqrt{4})^9 = 2 \cdot 2^9 = 2 \cdot 512 = 1024$ [1].

Stretch Challenge · +30 XP, +15 coins

Unit Boss — Everything at once

Simplify $\dfrac{\left(2 a^{1/2} b^{-1}\right)^4 \cdot a^0 b^3}{8 a^{-1} b^{-2}}$ leaving your answer with positive indices and fractional indices where unavoidable. Then state which laws you used at each stage. Finally, comment: is the result a polynomial in $a$ and $b$?

Reveal solution

Bracket (power of a product): $\left(2 a^{1/2} b^{-1}\right)^4 = 2^4 \cdot a^{1/2 \times 4} \cdot b^{-1 \times 4} = 16 a^2 b^{-4}$. Zero index: $a^0 = 1$. Numerator product: $16 a^2 b^{-4} \cdot b^3 = 16 a^2 b^{-1}$. Quotient: $\dfrac{16 a^2 b^{-1}}{8 a^{-1} b^{-2}} = 2 \cdot a^{2 - (-1)} \cdot b^{-1 - (-2)} = 2 a^3 b$. The answer is $2 a^3 b$ — yes, this is a polynomial (no negative or fractional indices remain).

Quick Review — The whole unit

Product

$x^m x^n = x^{m+n}$

Quotient

$x^m / x^n = x^{m-n}$

Power

$(x^m)^n = x^{mn}$

Zero

$x^0 = 1$

Negative

$x^{-n} = 1/x^n$

Fractional

$x^{m/n} = \sqrt[n]{x^m}$

Sci. notation

$a \times 10^n$, $1 \le a < 10$

3 sig fig

Standard answer precision

Strategy

Powers $\to$ $\times$ $\to$ $\div$

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