3. Indices and Standard Form

Your Guide to Indices & Standard Form

🧠 What This Chapter Is Really About

Indices & Standard Form let us write very large or very small numbers concisely and model exponential growth in real-life contexts. Why you care: from scientific notation in data to bank interest calculations, these tools 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 📌 In E-Math we compute answers directly; in A-Math we explore derivations, proofs & deeper connections.

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🔹 3.1 Indices

🧠 Key Concepts

Exponents compress repeated multiplication: $a^n=\underset{n\text{ times}}{\underbrace{a\times a\times \cdots \times a}}.$

🏷️ Definitions & Terminology

• Base ($a$): factor multiplied each time
• Index ($n$): count of multiplications

📐 Understanding the Parameters

• Larger $n$ grows values rapidly if $a>1$; if $0<a<1$, larger $n$ makes values shrink.

✏️ Worked Examples

Repeated multiplication Question: Express $2\times 2\times 2\times 2$ as an exponent.

• Step 1 (from question): Identify key info – factors are $2,2,2,2$.
• Step 2 (rule): Apply: write exponent form: $2^4$.
• Step 3 (interpretation): Interpret: exponent shows count of multiplications.

Repeated multiplication variation Question: Express $5\times 5\times 5$ as an exponent.

• Step 1 (from question): Identify key info – factors are $5,5,5$.
• Step 2 (rule): Apply: write exponent form: $5^3$.
• Step 3 (interpretation): Interpret: compact form highlights repeated multiplication.

Base of 1 Question: Simplify $1^n$ for any index $n$.

• Step 1 (from question): Identify key info – base 1 repeated $n$ times.
• Step 2 (rule): Apply: base-1 rule: $1^n=1$.
• Step 3 (interpretation): Interpret: result always 1 regardless of index.

🌐 Real-world Link

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

🤔 Check Yourself

1. Write $3\times 3\times 3\times 3\times 3$ as an exponent.
2. What is $1^n$ for any index $n$?

⚠️ Common Mistakes

• Confusing the number of factors with the base value.

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🔹 3.2 Laws of Indices

🧠 Key Concepts

Laws let you simplify exponent expressions without expanding: $a^m\times a^n=a^{m+n},\frac{a^m}{a^n}=a^{m-n},(a^m{)}^n=a^{mn},(ab{)}^n=a^n{b}^n.$

🏷️ Definitions & Terminology

• Product law: $a^m\times a^n=a^{m+n}$
• Quotient law: $a^m/a^n=a^{m-n}$
• Power law: $(a^m{)}^n=a^{mn}$
• Distribution law: $(ab{)}^n=a^n{b}^n$

📐 Understanding the Parameters

• Add/subtract laws apply only when bases match.

✏️ Worked Examples

Product law Question: Simplify $3^2\times 3^4$.

• Step 1 (from question): Identify key info – same base 3.
• Step 2 (rule): Apply: add exponents: $2+4=6$.
• Step 3 (interpretation): Interpret: result $3^6$.

Quotient law Question: Simplify $5^3/5^1$.

• Step 1 (from question): Identify key info – same base 5.
• Step 2 (rule): Apply: subtract exponents: $3-1=2$.
• Step 3 (interpretation): Interpret: result $5^2=25$.

Power law Question: Simplify $(2^3{)}^4$.

• Step 1 (from question): Identify key info – inner exponent 3, outer exponent 4.
• Step 2 (rule): Apply: multiply exponents: $3\times 4=12$.
• Step 3 (interpretation): Interpret: result $2^{12}=4096$.

Distribution law Question: Simplify $(2\times 3{)}^3$.

• Step 1 (from question): Identify key info – product $2\times 3$ raised to 3.
• Step 2 (rule): Apply: $2^3\times 3^3=8\times 27$.
• Step 3 (interpretation): Interpret: result $216$.

🌐 Real-world Link

Scaling dimensions in models or maps use exponent laws for area & volume.

🤔 Check Yourself

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

⚠️ Common Mistakes

• Applying exponent laws to different bases.

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🔹 3.3 Zero & Negative Indices

🧠 Key Concepts

Extending exponents to zero and negative values unifies reciprocals and unity: $a^0=1,a^{-n}=\frac{1}{a^n}.$

🏷️ Definitions & Terminology

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

📐 Understanding the Parameters

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

✏️ Worked Examples

Zero index Question: Simplify $5^0$.

• Step 1 (from question): Identify key info – base 5, exponent 0.
• Step 2 (rule): Apply: zero exponent rule: $5^0=1$.
• Step 3 (interpretation): Interpret: result is unity.

Negative index Question: Simplify $2^{-3}$.

• Step 1 (from question): Identify key info – base 2, exponent -3.
• Step 2 (rule): Apply: rewrite as reciprocal: $1/2^3$.
• Step 3 (interpretation): Interpret: compute: $1/8$.

🌐 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

• Assuming $a^0=0$ instead of 1.

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🔹 3.4 Rational Indices

🧠 Key Concepts

Fractional exponents represent roots: $a^{m/n}=\sqrt[n]{a^m}=(\sqrt[n]{a}{)}^m.$

🏷️ Definitions & Terminology

• $a^{1/2}=\sqrt{a}$
• $a^{1/3}=\sqrt[3]{a}$
• $a^{m/n}=(\sqrt[n]{a}{)}^m$

📐 Understanding the Parameters

• Denominator $n$ = root degree; numerator $m$ = power after root.

✏️ Worked Examples

Square-root exponent Question: Simplify $9^{1/2}$.

• Step 1 (from question): Identify key info – base 9, exponent $\frac{1}{2}$.
• Step 2 (rule): Apply: square root: $\sqrt{9}$.
• Step 3 (interpretation): Interpret: result $3$.

Cube-power fractional exponent Question: Simplify $27^{2/3}$.

• Step 1 (from question): Identify key info – base 27, exponent $\frac{2}{3}$.
• Step 2 (rule): Apply: cube root then square: $\sqrt[3]{27}=3$, then $3^2=9$.
• Step 3 (interpretation): Interpret: result $9$.

Fourth-root exponent Question: Simplify $16^{3/4}$.

• Step 1 (from question): Identify key info – base 16, exponent $\frac{3}{4}$.
• Step 2 (rule): Apply: fourth root then cube: $\sqrt[4]{16}=2$, then $2^3=8$.
• Step 3 (interpretation): Interpret: result $8$.

🌐 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

Models money growth with periodic compounding: $A=P(1+\frac{r}{n}{)}^{nt}.$

🏷️ Definitions & Terminology

• $P$: principal
• $r$: annual rate (decimal)
• $n$: 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 Question: Calculate $A$ for $P=1000$, $r=5\%$, $n=1$, $t=3$.

• Step 1 (from question): Identify key info – principal 1000, rate 5%, periods 1, time 3 years.
• Step 2 (rule): Apply: $A=1000(1+0.05{)}^3$.
• Step 3 (interpretation): Interpret: compute $1000\times 1.157625=1157.63$.

Semiannual compounding Question: Calculate $A$ for $P=500$, $r=4\%$, $n=2$, $t=2$.

• Step 1 (from question): Identify key info – principal 500, rate 4%, periods 2, time 2 years.
• Step 2 (rule): Apply: rate per period $0.04/2=0.02$, $A=500(1.02{)}^4$.
• Step 3 (interpretation): Interpret: compute $\approx 520.40$.

Simple vs compound Question: Compare simple vs compound interest for $P=1000$, $r=5\%$, $t=2$.

• Step 1 (from question): Identify key info – principal 1000, rate 5%, time 2 years.
• Step 2 (rule): Apply: simple: $1000(1+0.05\times 2)=1100$; compound: $1000(1.05{)}^2=1102.50$.
• Step 3 (interpretation): Interpret: compound yields more due to interest-on-interest.

🌐 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

Express numbers as $a\times 10^n$ 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 Question: Convert $0.00047$ to standard form.

• Step 1 (from question): Identify key info – decimal number 0.00047.
• Step 2 (rule): Apply: move decimal 4 places right, obtaining $4.7$, so exponent $n=-4$.
• Step 3 (interpretation): Interpret: write $4.7\times 10^{-4}$.

Large number to standard form Question: Convert $3,200,000$ to standard form.

• Step 1 (from question): Identify key info – number 3,200,000.
• Step 2 (rule): Apply: move decimal 6 places left, obtaining $3.2$, so exponent $n=6$.
• Step 3 (interpretation): Interpret: write $3.2\times 10^6$.

Standard form to decimal Question: Convert $5.6\times 10^{-3}$ to decimal.

• Step 1 (from question): Identify key info – coefficient 5.6, exponent -3.
• Step 2 (rule): Apply: move decimal 3 places left.
• Step 3 (interpretation): Interpret: write $0.0056$.

🌐 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 $n$.