3. Indices and Standard Form

Your Guide to Indices & Standard Form

🧠 What This Chapter Is Really About

With Indices & Standard Form, we can represent very large or very small numbers in a simple way and model exponential growth in real-life contexts.

From scientific notation in data to bank interest calculations, exponents are what power everyday tech.

• using exponents to simplify repeated multiplication
• applying and combining laws of indices
• extending exponents to zero, negative, and fractional powers
• modelling compound interest growth
• writing numbers uniformly in standard form

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🔹 3.1 Indices

🧠 Key Concept

Indices represent repeated multiplication:

$a^n=\underset{n\text{ times}}{\underbrace{a\times a\times \cdots \times a}}.$

🏷️ Definitions & Terminology

Informally, $a^n$ can be seen as ${\text{Base}}^{\text{Index}}$.

• Base ($a$): number being repeated
• Index ($n$): no. of times base appears

Worked Examples

Repeated multiplication

Question: Express $2\times 2\times 2\times 2$ in index form.

• Base (number being repeated) = 2.
• Index (no. of times base appears) = 4
• Answer: $2^4$

🌐 Real-world Link

Memory sizes in computing use exponents (e.g. $2^{10}=1024$ bytes = $1$ kilobyte).

⚠️ Common Mistakes

• Confusing base and index!

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🔹 3.2 Laws of Indices

🧠 Key Concepts

Un-simplified index expressions can look very complicated. How do we break them down?

🏷️ 5 Laws of Indices

• Same base, different power $\begin{array}{rl}\text{Law 1 (index addition): }& a^m\times a^n=a^{m+n}\\ \text{Law 2 (index subtraction): }& a^m÷a^n=a^{m-n}\\ \text{Law 3 (index multiplication): }& (a^m{)}^n=a^{mn}\\ \text{Law 3 (multiplied by rational index): }& (a^m{)}^{\frac{1}{n}}=(a^{\frac{1}{n}}{)}^m=a^{\frac{m}{n}}\end{array}$
• Same power, different base $\begin{array}{rl}\text{Law 4 (base multiplication): }& a^n\times b^n=(a\times b{)}^n\\ \text{Law 5 (base division): }& a^n÷b^n=(\frac{a}{b}{)}^n=\frac{a^n}{b^n}\end{array}$

📐 Understanding the Parameters

• Either base or power must match, otherwise we can’t use laws!

Worked Examples

Easy

1. Simplify $3^2\times 3^4$.

• This is law 1 $⟹$ Add the indices!
• $3^2\times 3^4=3^{2+4}=3^6$.
• Answer: $3^6$

2. Simplify $5^7/5^2$.

• Law 2 $⟹$ Subtract the indices!
• $$\begin{gathered} 5^7÷5^2=5^{7-2} \\ =5^5 \end{gathered}$$.
• Answer: $5^5$

3. Simplify $(2^3{)}^4$.

• Law 3 $⟹$ Multiply the indices!
• • $(2^3{)}^4=2^{3\times 4}=2^{12}$.
• Answer: $2^{12}$

4. Simplify $(2\times 3{)}^3$.

• Law 4 $⟹$ Multiply the bases!
• $2^3\times 3^3=6^3=216$.
• Answer: $216$

5. Simplify $14^{13}÷7^{13}$.

• Law 5 $⟹$ Divide the bases!
• $14^{13}÷7^{13}=2^{13}$.
• Answer: $2^{13}=8192$

Medium

6. Simplify $(-15y^9)÷5y^4$.

• Law 2 $⟹$ Divide the indices!
• Tip: You can handle the numbers separately!
• $\frac{-15y^9}{5y^4}=\frac{-15}{5}\times \frac{y^9}{y^4}=-3y^5$

7. Simplify $(4k^6{)}^3÷(-2k^3{)}^3$.

• Law 3 and Law 5 can both be applied. Which first?
• Law 3 first $\begin{array}{rl}(4k^6{)}^3÷(-2k^3{)}^3& =(4{)}^3{k}^{18}÷(-2{)}^3{k}^9\text{ (Law 3 - Multiply indices!)}\\ & =64k^{18}÷-8k^9\\ & =\frac{64}{-8}\times \frac{k^{18}}{k^9}\\ & =-8k^9\text{ (Law 2 - Subtract the indices!)}\end{array}$
• Law 5 first $\begin{array}{rl}(4k^6{)}^3÷(-2k^3{)}^3& =(\frac{4k^6}{-2k^3}{)}^3\text{ (Law 5 - Divide base!)}\\ & =(\frac{4}{-2}\times \frac{k^6}{k^3}{)}^3\\ & =(-2k^3{)}^3\text{ (Law 2 - Subtract indices!)}\\ & =-8k^9\text{ (Law 3 - Multiply indices!)}\end{array}$
• In this case, Law 5 is easier (smaller numbers), but you can take your pick!

🤔 Check Yourself

1. Simplify $2^5\times 2^{-2}$.
2. Simplify $(4^2{)}^3$.

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🔹 3.3 Zero & Negative Indices

🧠 Key Concepts

What happens when the index is 0, or if it is negative?

$a^0=1\text{ (Zero index)},a^{-n}=\frac{1}{a^n}\text{ (Negative index)}$

🏷️ Definitions & Terminology

• Zero index: $a^0=1$ (for $a\ne 0$)
• Negative index: $a^{-n}=\frac{1}{a^n}$

📐 Understanding the Parameters

• Zero exponent ignores the base (except 0); negative flips expression to its reciprocal.

Worked Examples

Easy

1. Simplify $5^0$.

• Any non-zero number to the power of 0 is 1!
• Answer: $5^0=1$

2. Simplify $4^{-3}$.

• Negative index $⟹$ Turn it into a fraction!
• Answer: $4^{-3}=\frac{1}{4^3}$

3. Express $\frac{1}{16}$ in index form.

• It works the other way round too!
• Answer: $\frac{1}{16}=\frac{1}{2^4}=2^{-4}$
• Alternate answer: $\frac{1}{16}=\frac{1}{4^2}=4^{-2}$

🌐 Real-world Link

pH scale in chemistry uses negative exponents for hydrogen ion concentration.

🤔 Check Yourself

1. What is $a^0$ for any nonzero $a$?
2. Express $1/16$ as an exponent of 2.

⚠️ Common Mistakes

• Why is $a^0=0$ wrong?

• Why is $-a^3=\frac{1}{a^3}$ wrong?

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🔹 3.4 Rational Indices

🧠 Key Concepts

Rational numbers: Numbers that can be expressed as fractions $⟹$ $\frac{m}{n}$ is a rational number.

Rational indices $⟹$ Index is a rational number (i.e. $a^{\frac{m}{n}}$)

$a^{\frac{m}{n}}=\sqrt[n]{a^m}=(\sqrt[n]{a}{)}^m.$

🏷️ Definitions

• Radical sign: $\surd$
• Square Root: $a^{\frac{1}{2}}=\sqrt{a}$
• Cube Root: $a^{\frac{1}{3}}=\sqrt[3]{a}$

🔑 Different Forms of Rational Indices

$a^{\frac{m}{n}}=\sqrt[n]{a^m}=(\sqrt[n]{a}{)}^m$

• These 3 forms mean the same thing!
• 1st form: $a^{\frac{m}{n}}$ is useful for combining with other Laws of Indices
  • e.g. $(x^{\frac{2}{5}}{)}^5=x^2$
• 2nd form: $\sqrt[n]{a^m}$ is useful if $a^m$ can be simplified mentally
  • e.g. $\sqrt[3]{8^2}=\sqrt[3]{64}=\sqrt[3]{4^3}=4$
• 3rd form: $(\sqrt[n]{a}{)}^m$ is useful if $a$ is a big number but can be simplified using the nth root
  • e.g. $625^{\frac{3}{4}}=(\sqrt[4]{625}{)}^3=(\sqrt[4]{5^4}{)}^3=5^3=125$
  • Tip: Don’t be intimidated by big numbers! Use prime factorisation!
    • $625=5\times 125=5^2\times 25=5^3\times 5=5^4$

Worked Examples

Square-root exponent

• Simplify $9^{1/2}$.
  • Square root exponent: $9^{\frac{1}{2}}=\sqrt{9}=3$
  • Answer: 3

Rational index

1. Simplify $27^{2/3}$.

• Step 1 (try rational index forms): $27^{\frac{2}{3}}=\sqrt[3]{27^2}=(\sqrt[3]{27}{)}^2$

• Step 2: Which looks easier to calculate mentally?

  • Not $27^{\frac{2}{3}}$ - Can’t calculate directly
  • Not $\sqrt[3]{27^2}$ - What’s $27^2$? It’s 729 but hard to do mentally
  • $(\sqrt[3]{27}{)}^2$ is easiest!

• Step 3: Simplify $(\sqrt[3]{27}{)}^2$

  • Break down the number inside: $27=3\times 9=3^2\times 3=3^3$
  • Sub back in: $(\sqrt[3]{27}{)}^2=(\sqrt[3]{3^3}{)}^2=3^2=9$

• Answer: 9

2. Simplify $16^{3/4}$.

• Step 1 (try rational index form): $16^{\frac{3}{4}}=\sqrt[4]{16^3}=(\sqrt[4]{16}{)}^3$
• Step 2: Which looks easier to calculate mentally?
  • Not $16^{\frac{3}{4}}$ - Can’t calculate directly
  • Not $\sqrt[4]{16^3}$ - What’s $16^3$? It’s $4096$ but hard to do mentally
  • $(\sqrt[4]{16}{)}^3$ is easiest!
• Step 3: Simplify $(\sqrt[4]{16}{)}^3$
  • Break down the number inside: $16=2\times 8=2^2\times 4=2^3\times 2=2^4$
  • Sub back in: $(\sqrt[4]{16}{)}^3-(\sqrt[4]{2^4}{)}^3=2^3$

🌐 Real-world Link

Fractional exponents appear in physics wave equations and growth models.

🤔 Check Yourself

1. Evaluate $8^{2/3}$.
2. Write $\sqrt[5]{32}$ as an exponent.

⚠️ Common Mistakes

• Swapping numerator and denominator in the exponent.

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🔹 3.5 Compound Interest

🧠 Key Concepts

Compound interest is used primarily for finances, being used for: investment growth projection, monthly loan calculation, monthly savings interest calculation, credit card bill calculation and more:

➕ Simple Interest - Formula

$I_{simple}=\frac{PRT}{100}$

• $I_{simple}$: simple interest amount
• $P$: principal
• $R\%$: annual interest rate
• $T$: time in years

✖️ Compound Interest - Formula

$\begin{array}{rl}A& =P(1+\frac{r}{100n}{)}^{nt}.\\ I_{compound}& =A-P\\ & =P(1+\frac{r}{100n}{)}^{nt}-P\end{array}$

• $A$: accumulated amount
• $I_{compound}$: compound interest amount
• $P$: principal
• $r$: annual interest rate
• $n$: compounding periods per year
• $t$: time in years

📐 Understanding the Parameters

• Higher $n$ yields slightly more growth; rate $r$ and time $t$ scale growth.

Worked Examples

Annual compounding

Calculate $A$ for $P=1000$, $r=5\%$, $n=1$, $t=3$. $\begin{array}{rl}A& =P(1+\frac{r}{100}{)}^t\text{ (simplified compounding formula)}\\ & =1000(1+\frac{5}{100}{)}^3\text{ (sub in P, r, t)}\\ & =1157.625\\ & \approx \$1157.63\end{array}$ Answer: $1157.63.

Semiannual compounding

Calculate $A$ for $P=500$, $r=4\%$, $n=2$, $t=2$. \begin{aligned} A &= P(1 + \frac{r}{100n})^t) \text{ (compounding formula)} \\ &= 500(1 + \frac{4}{100 \times 2})^2 \text{ (sub in P, r, t, n)} \\ &= $520.20 \end{aligned} Answer: $520.20

Simple vs compound

Compare simple vs annual compound interest for $P=10000$, $r=5\%$, $t=2$.

Simple interest \begin{aligned} I_{simple} &= \frac{PRT}{100} \text{ (simple interest formula)} \\ &= \frac{10000 \times 5 \times 2}{100} \\ &= $1000 \end{aligned} Simple interest amount: $1000

Compound interest \begin{aligned} A &= P(1 + \frac{r}{100})^t \\ &= 1000(1 + \frac{5}{100})^2 \\ &= 11025 \\ I &= A - P = 11025 - 10000 = $1025 \end{aligned} Compound interest amount: $1025

Comparison Compound interest yields slightly more than simple interest over 2 years. What happens if we compound this over longer?

• 2 years: $1000vs$1025 - Difference: $25
• 5 years: $2500vs$2762.82 - Difference: $262.82
• 10 years: $5000vs$6288.95 - Difference: $1288.95
• 20 years: $10000vs$16532.98 - Difference: $6532.98
• 30 years: $15000vs$33219.42 - Difference: $18219.42

We see that compound interest grows exponentially given time. What happens if you change the P, R or T?

🌐 Real-world Link

Savings accounts, loans, and mortgages use compound interest.

🤔 Check Yourself

1. Write formula when $n=4$ and $r=6\%$.
2. How adjust for simple interest?

⚠️ Common Mistakes

• Using simple interest formula when compounding applies.

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🔹 3.6 Standard Form

🧠 Key Concepts

Standard form aims to compress really large or really small numbers in a simple way! Numbers look like $a\times 10^n$, where $1\le a<10$ to handle extremes.

🏷️ Definitions & Terminology

• Standard form: $a\times 10^n$ with $1\le a<10$.

📐 Understanding the Parameters

• Positive $n$ for large numbers; negative $n$ for decimals.

Worked Examples

Decimal to standard form

Convert $0.00047$ to standard form.

• Step 1: Count decimal places
  • We want it to become $4.7\times 10^n$. How many places do we need to shift the decimal place?
  • 4 decimal places right
• Step 2: Convert to standard form
  • Since we shift right, $n$ in $10^n$ should be negative.
  • $0.00047=4.7\times 10^{-4}$
• Answer: $4.7\times 10^{-4}$

Large number to standard form

Convert $3,200,000$ to standard form.

• Step 1: Count decimal places
  • We want it to become $3.2\times 10^n$. How many places do we need to shift the decimal place?
  • 6 decimal places left
• Step 2: Convert to standard form
  • Since we shift left, $n$ in $10^n$ should be positive.
  • $3,200,000=3.2\times 10^6$
• Answer: $3.2\times 10^6$

Standard form to decimal

Convert $5.6\times 10^{-3}$ to decimal.

• Step 1: Identify decimal places to shift
  • Since $n=-3$ (n is negative), the decimal place has already been shifted right 3 times.
  • We need to reverse this.
• Step 2: Convert to decimal
  • We shift the decimal place left 3 times!
  • $5.6\times 10^{-3}=0.0056$
• Answer: $0.0056$.

🌐 Real-world Link

Scientific data in physics and astronomy relies on standard form.

🤔 Check Yourself

1. Write $45000$ in standard form.
2. Convert $7.89\times 10^2$ to decimal.

⚠️ Common Mistakes

• Choosing $a$ outside $[1,10)$.

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✅ Quick Strategy 🎯

1. Identify task: write, simplify or apply indices.
2. Match base or operation; use correct law or definition.
3. For zero/negative indices, apply unity or reciprocal rules.
4. For fractional indices, root then power.
5. For interest, plug into $A=P(1+\frac{r}{n}{)}^{nt}$.
6. For standard form, shift decimal so $1\le a<10$, count shifts for $n$.