Comprehensive Study Guide: Indices and Surds

Maths in Context: Indices and Piano Note Frequencies

• An octave is a musical interval consisting of 12 consecutive semitones. When moving up an octave from a starting note C, the frequency doubles.

• For a piano, multiplying the lowest C frequency by $2^1, 2^2, 2^3, 2^4$, etc., calculates all its higher C frequencies. Multiplying the highest C frequency by $2^{-1}, 2^{-2}, 2^{-3}, 2^{-4}$, etc., gives all lower C frequencies.

• Within an octave, the frequency increases from one semitone to the next by a factor of $2^{\frac{1}{12}}$ because:

$(2^{\frac{1}{12}})^{12} = 2^{\frac{1}{12} \times 12} = 2^1 = 2$

• Consequently, the 11 notes above a starting C in an octave are found by multiplying the C frequency by:

$2^{\frac{1}{12}}, (2^{\frac{1}{12}})^2, (2^{\frac{1}{12}})^3, \dots, (2^{\frac{1}{12}})^{10}, (2^{\frac{1}{12}})^{11}$

• Frequencies starting from Middle C ($261.6\,\text{Hz}$):

  • Middle C (1st note): $261.6\,\text{Hz}$

  • C sharp (2nd note): $261.6 \times 2^{\frac{1}{12}} = 277.2\,\text{Hz}$

  • D (3rd note): $261.6 \times (2^{\frac{1}{12}})^2 = 293.6\,\text{Hz}$

  • B (12th note): $261.6 \times (2^{\frac{1}{12}})^{11} = 408.9\,\text{Hz}$

  • Next higher C: $261.6 \times (2^{\frac{1}{12}})^{12} = 261.6 \times 2 = 523.2\,\text{Hz}$

• Tuning concert pitch: Standard concert pitch tunes A above middle C to $440\,\text{Hz}$. Each note above A is $\times 2^{\frac{1}{12}}$ of the frequency of the note below it; each note below A is $\times 2^{-\frac{1}{12}}$ of the frequency of the note above it. Since Middle C is 9 semitones below A, its frequency is calculated as:

$440 \times (2^{-\frac{1}{12}})^9 = 261.6\,\text{Hz}$

Index Notation (Consolidating)

• Indices (plural of index) provide a concise shorthand notation for repeated multiplication of the same factor.

• Key Components of a Power:

  • Power: An expression written in index form, such as $a^n$.

  • Base ($a$): The factor being repeatedly multiplied.

  • Index / Exponent / Power ($n$): The number of times the base factor is written.

  • Basic Numeral: The evaluated single number value of a power (e.g., in $2^5 = 32$, $32$ is the basic numeral).

  • Fundamental Rule: $a^1 = a$ (e.g., $5^1 = 5$).

• Representation Comparison:

  • Expanded Form: $2 \times 2 \times 2 \times 2 \times 2$

  • Index Form: $2^5$

  • Basic Numeral: $32$

  • Pronumeral Expanded Form: $x \times x \times x \times x$

  • Pronumeral Index Form: $x^4$

• Prime Factorisation:

  • Prime factorisation involves expressing a composite number as a product of its prime factors in index notation.

  • Example (Factor Tree for $108$):

$108 = 2 \times 54 = 2 \times 2 \times 27 = 2 \times 2 \times 3 \times 9 = 2 \times 2 \times 3 \times 3 \times 3 = 2^2 \times 3^3$

• Real-World Example (Mass Decay):

  • The mass of a $100\,\text{kg}$ limestone block decreasing by $2\%$ per year over $20$ years is calculated as:

$\text{Mass} = 100 \times (0.98)^{20}\,\text{kg}$

• Professional Applications:

  • Index laws are utilized across numerous fields including by scientists, chemists, radiographers, financial planners, economists, actuaries, astronomers, geologists, computer programmers, and medical, electrical, sound, and aerospace engineers.

• Worked Examples:

  • Example 1 (Expanded Form):

  • a) $a^3 = a \times a \times a$

  • b) $(xy)^4 = xy \times xy \times xy \times xy$

  • c) $2a^3 b^2 = 2 \times a \times a \times a \times b \times b$

  • Example 2 (Expanding and Evaluating):

  • a) $5^3 = 5 \times 5 \times 5 = 125$

  • b) $(-2)^5 = (-2) \times (-2) \times (-2) \times (-2) \times (-2) = -32$

  • c) $(\frac{2}{5})^3 = \frac{2}{5} \times \frac{2}{5} \times \frac{2}{5} = \frac{8}{125}$

  • Example 3 (Index Form):

  • a) $6 \times x \times x \times x \times x = 6x^4$

  • b) $\frac{3}{7} \times \frac{3}{7} \times \frac{4}{5} \times \frac{4}{5} \times \frac{4}{5} = (\frac{3}{7})^2 \times (\frac{4}{5})^3$

  • c) $8 \times a \times a \times 8 \times b \times b \times a \times b = 8 \times 8 \times a \times a \times a \times b \times b \times b = 8^2 a^3 b^3$

  • Example 4 (Prime Factor Form):

  • Express $48$ as a product of prime factors in index form:

$48 = 8 \times 6 = (2 \times 4) \times (2 \times 3) = 2 \times (2 \times 2) \times 2 \times 3 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3$

• Key Principles for Powers of Negative Numbers:

  • Even powers of negative base numbers produce positive results:

$(-3)^2 = 9, \quad (-4)^2 = 16$

• Odd powers of negative base numbers produce negative results:

$(-2)^3 = -8, \quad (-2)^5 = -32$

• Distinguish carefully between brackets and leading negative signs:

$(-3)^2 = 9 \quad \text{whereas} \quad -(3)^2 = -9$

• Decimal Powers as Fractions:

  • Converting decimals to fractions simplifies mental power calculations:

$(0.5)^4 = (\frac{1}{2})^4 = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{16}$

• Highest Common Factor (HCF) and Lowest Common Multiple (LCM) from Prime Factorisation:

  • HCF: Product of all common prime factors raised to their smallest power.

  • LCM: Product of all different prime factors raised to their highest power.

  • Example for $12$ and $30$:

$12 = 2^2 \times 3, \quad 30 = 2 \times 3 \times 5$

$\text{Common prime factors: } 2 \text{ and } 3 \implies \text{HCF} = 2^1 \times 3^1 = 6$

$\text{Highest powers: } 2^2, 3^1, 5^1 \implies \text{LCM} = 2^2 \times 3 \times 5 = 60$

Index Laws for Multiplication and Division

• An index law (or identity) is an algebraic equation that holds true for all possible values of the variables involved.

• Index Law for Multiplication:

  • When multiplying terms with the same base, keep the base and add the powers:

$a^m \times a^n = a^{m+n}$

• Derivation via expanded form:

$a^m \times a^n = \underbrace{(a \times a \times \dots \times a)}_{m \text{ factors}} \times \underbrace{(a \times a \times \dots \times a)}_{n \text{ factors}} = \underbrace{a \times a \times \dots \times a}_{m+n \text{ factors}} = a^{m+n}$

• Index Law for Division:

  • When dividing terms with the same base, keep the base and subtract the powers:

$a^m \div a^n = \frac{a^m}{a^n} = a^{m-n}$

• Derivation via cancelling factors:

$a^m \div a^n = \frac{\overbrace{a \times a \times \dots \times a}^{m \text{ factors}}}{\underbrace{a \times a \times \dots \times a}_{n \text{ factors}}} = a^{m-n}$

• Real-World Modeling Application:

  • Bacteria population after $n$ generations: $P = 2^n$

  • Projected Australian population in millions $n$ years from 2018: $P = 25 \times (1.04)^n$

• Worked Examples:

  • Example 5 (Numerical Bases):

  • a) $3^6 \times 3^4 = 3^{6+4} = 3^{10}$

  • b) $7^9 \div 7^5 = 7^{9-5} = 7^4$

  • Example 6 (Algebraic Multiplication):

  • a) $x^4 \times x^5 = x^{4+5} = x^9$

  • b) $x^3 y^4 \times x^2 y = x^{3+2} y^{4+1} = x^5 y^5$

  • c) $3m^4 \times 2m^5 = (3 \times 2) \times m^{4+5} = 6m^9$

  • Example 7 (Algebraic Division):

  • a) $x^{10} \div x^2 = x^{10-2} = x^8$

  • b) $\frac{8a^6 b^3}{12a^2 b^2} = \frac{8}{12} a^{6-2} b^{3-2} = \frac{2}{3} a^4 b = \frac{2a^4 b}{3}$

  • Example 8 (Combining Multiplication and Division):

  • a) $x^2 \times x^3 \div x^4 = x^{2+3} \div x^4 = x^5 \div x^4 = x^{5-4} = x^1 = x$

  • b) $\frac{2a^3 b \times 8a^2 b^3}{4a^4 b^2} = \frac{16 a^{3+2} b^{1+3}}{4a^4 b^2} = \frac{16 a^5 b^4}{4a^4 b^2} = 4 a^{5-4} b^{4-2} = 4ab^2$

Index Laws for Power of a Power and the Zero Index

• Index Law for Power of a Power:

  • When raising a term in index form to another power, retain the base and multiply the indices:

$(a^m)^n = a^{m \times n} = a^{mn}$

• Derivation:

$(a^m)^n = \underbrace{a^m \times a^m \times \dots \times a^m}_{n \text{ factors of } a^m} = a^{\overbrace{m + m + \dots + m}^{n \text{ times}}} = a^{mn}$

• The Zero Index Rule:

  • Any non-zero expression raised to the power of zero equals 1:

$a^0 = 1, \quad \text{where } a \neq 0$

• Proof using division law:

$\frac{a^m}{a^m} = 1$

$\text{Using division law: } \frac{a^m}{a^m} = a^{m-m} = a^0$

$\implies a^0 = 1$

• Crucial Bracketing Distinctions with Zero Powers:

  • $(2a)^0 = 1$ (the power applies to the entire product $2a$)

  • $2a^0 = 2 \times 1 = 2$ (the power applies only to $a$)

  • $(-3)^0 = 1$

  • $-(5x)^0 = -1$

  • $2y^0 - (3y)^0 = 2(1) - 1 = 2 - 1 = 1$

• Worked Examples:

  • Example 9 (Power of a Power):

  • a) $(x^5)^4 = x^{5 \times 4} = x^{20}$

  • b) $3(y^5)^2 = 3y^{5 \times 2} = 3y^{10}$

  • Example 10 (Zero Index):

  • a) $(-3)^0 = 1$

  • b) $-(5x)^0 = -1$

  • c) $2y^0 - (3y)^0 = 2(1) - 1 = 1$

  • Example 11 (Combining Laws):

  • a) $(x^2)^3 \times (x^3)^5 = x^6 \times x^{15} = x^{6+15} = x^{21}$

  • b) $\frac{(m^3)^4}{m^7} = \frac{m^{12}}{m^7} = m^{12-7} = m^5$

  • c) $\frac{4x^2 \times 3x^3}{6x^5} = \frac{12x^{2+3}}{6x^5} = \frac{12x^5}{6x^5} = 2x^{5-5} = 2x^0 = 2 \times 1 = 2$

• Exponential Population Growth Example (Mt Burrow Rabbits):

  • Starting population in 2010 ($t = 0$): $100$

  • Growth model: $N = 100 \times 2^t$

  • At $t = 0$ ($2010$): $N = 100 \times 2^0 = 100$

  • At $t = 2$ ($2012$): $N = 100 \times 2^2 = 400$

  • At $t = 6$ ($2016$): $N = 100 \times 2^6 = 6400$

• Base Transformation Techniques:

  • Base numbers can be converted to prime bases using power of a power:

$8^2 = (2^3)^2 = 2^6, \quad 9^3 = (3^2)^3 = 3^6, \quad 32^3 = (2^5)^3 = 2^{15}, \quad 81^5 = (3^4)^5 = 3^{20}$

Index Laws for Brackets and Fractions

• Index Law for Brackets (Power of a Product):

  • When multiplying two or more terms inside brackets raised to a power $m$, raise each factor inside the brackets to the power $m$:

$(a \times b)^m = (ab)^m = a^m b^m$

• Derivation:

$(ab)^m = \underbrace{ab \times ab \times \dots \times ab}_{m \text{ factors}} = \underbrace{(a \times a \times \dots \times a)}_{m \text{ factors}} \times \underbrace{(b \times b \times \dots \times b)}_{m \text{ factors}} = a^m b^m$

• Index Law for Fractions (Power of a Quotient):

  • When dividing two terms inside brackets raised to a power $m$, raise both the numerator and denominator to the power $m$ ($b \neq 0$):

$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$

• Derivation:

$\left(\frac{a}{b}\right)^m = \underbrace{\frac{a}{b} \times \frac{a}{b} \times \dots \times \frac{a}{b}}_{m \text{ factors}} = \frac{\overbrace{a \times a \times \dots \times a}^{m \text{ factors}}}{\underbrace{b \times b \times \dots \times b}_{m \text{ factors}}} = \frac{a^m}{b^m}$

• Real-World Finance Application (Compound Interest):

  • An amount $\$P$ invested at $r\%$ p.a. for $n$ years grows to:

$A = P \left(1 + \frac{r}{100}\right)^n$

• Worked Examples:

  • Example 12 (Index Law for Brackets):

  • a) $(5b)^3 = 5^3 b^3 = 125b^3$

  • b) $(-2x^3 y)^4 = (-2)^4 (x^3)^4 y^4 = 16x^{12} y^4$

  • c) $4(c^2 d^3)^5 = 4(c^2)^5 (d^3)^5 = 4c^{10} d^{15}$

  • d) $3a^2 (2a^3)^4 = 3a^2 \times 2^4 (a^3)^4 = 3a^2 \times 16 a^{12} = 48a^{2+12} = 48a^{14}$

  • Example 13 (Index Law for Fractions):

  • a) $(\frac{6}{b})^3 = \frac{6^3}{b^3} = \frac{216}{b^3}$

  • b) $(\frac{-2a^2}{3bc^3})^4 = \frac{(-2)^4 (a^2)^4}{3^4 b^4 (c^3)^4} = \frac{16a^8}{81b^4 c^{12}}$

  • c) $(\frac{x^2 y^3}{c})^3 \times (\frac{xc}{y})^4 = \frac{x^6 y^9}{c^3} \times \frac{x^4 c^4}{y^4} = \frac{x^{10} y^9 c^4}{c^3 y^4} = x^{10} y^5 c$

Negative Indices

• Definition and Derivation:

  • A negative index indicates division or reciprocal relationship:

$a^{-m} = \frac{1}{a^m}$

• Derivation via division:

$\frac{a^2}{a^5} = a^{2-5} = a^{-3}$

$\text{Alternatively: } \frac{a^2}{a^5} = \frac{a \times a}{a \times a \times a \times a \times a} = \frac{1}{a^3} \implies a^{-3} = \frac{1}{a^3}$

• Core Negative Index Rules:

  • Negative index in denominator:

$\frac{1}{a^{-m}} = a^m$

• Reciprocal rule:

$a^{-1} = \frac{1}{a}$

• Fraction raised to negative power:

$\left(\frac{a}{b}\right)^{-1} = \frac{b}{a}$

$\left(\frac{a}{b}\right)^{-m} = \left(\frac{b}{a}\right)^m = \frac{b^m}{a^m}$

• Real-World Audio Engineering Application:

  • Inverse square law for sound intensity $I$ relative to distance $d$:

$I \propto \frac{1}{d^2} \implies I \propto d^{-2}$

• Worked Examples:

  • Example 14 (Expressing with Positive Indices):

  • a) $x^{-2} = \frac{1}{x^2}$

  • b) $3a^{-2} b^4 = \frac{3}{1} \times \frac{1}{a^2} \times \frac{b^4}{1} = \frac{3b^4}{a^2}$

  • Example 15 (Moving Negative Powers Across Fractions):

  • a) $\frac{1}{c^{-2}} = c^2$

  • b) $\frac{5}{x^3 y^{-4}} = \frac{5y^4}{x^3}$

  • c) $\frac{x^{-3}}{y^{-5}} = \frac{1}{x^3} \times \frac{y^5}{1} = \frac{y^5}{x^3}$

  • Example 16 (Evaluating without Calculator):

  • a) $3^{-4} = \frac{1}{3^4} = \frac{1}{81}$

  • b) $\frac{-5}{3^{-2}} = -5 \times 3^2 = -5 \times 9 = -45$

  • c) $(\frac{2}{3})^{-4} = \frac{2^{-4}}{3^{-4}} = \frac{3^4}{2^4} = \frac{81}{16}$

Scientific Notation

• General Form of Scientific Notation (Standard Form):

$a \times 10^m, \quad \text{where } 1 \le a < 10 \text{ (or } -10 < a \le -1\text{) and } m \in \mathbb{Z}$

• Sign of the Power $m$:

  • Large numbers ($\ge 10$): Use positive powers of $10$.

  • Example: $38\,\text{million years} = 38000000 = 3.8 \times 10^7$

  • Small numbers ($< 1$): Use negative powers of $10$.

  • Example: $417\,\text{nanoseconds} = 0.000000417 = 4.17 \times 10^{-7}$

• Metric Prefixes and Corresponding Powers of 10:

  • Tera ($\text{T}$): $10^{12}$ (e.g., $1\,\text{terabyte} = 10^{12}\,\text{bytes}$)

  • Giga ($\text{G}$): $10^9$ (e.g., $16\,\text{gigabytes} = 1.6 \times 10^{10}\,\text{bytes}$)

  • Mega ($\text{M}$): $10^6$ (e.g., $1\,\text{megalitre} = 10^6\,\text{litres}$)

  • Kilo ($\text{k}$): $10^3$ (e.g., $2178\,\text{km} = 2.178 \times 10^6\,\text{m}$)

  • Milli ($\text{m}$): $10^{-3}$ (e.g., $109\,\text{ms} = 0.109\,\text{s} = 1.09 \times 10^{-1}\,\text{s}$)

  • Micro ($\mu$): $10^{-6}$ (e.g., $27\,\text{microns} = 2.7 \times 10^{-5}\,\text{m}$)

  • Nano ($\text{n}$): $10^{-9}$ (e.g., $54\,\text{nanoseconds} = 5.4 \times 10^{-8}\,\text{s}$)

  • Pico ($\text{p}$): $10^{-12}$ (e.g., $18\,\text{picograms} = 1.8 \times 10^{-11}\,\text{g}$)

• Real-World Context Examples:

  • Concrete in Hoover Dam: $3400000\,\text{m}^3 = 3.4 \times 10^6\,\text{m}^3$

  • Mass of water molecule: $0.0000000000000000000000299\,\text{g} = 2.99 \times 10^{-23}\,\text{g}$

  • Targeted nanobots: $3 \times 10^{-6}\,\text{m}$ long with $1 \times 10^{-9}\,\text{m}$ components.

• Worked Examples:

  • Example 17 (Converting to Scientific Notation):

  • a) $4500000 = 4.5 \times 10^6$ (decimal shifted 6 places left)

  • b) $0.0000004 = 4 \times 10^{-7}$ (decimal shifted 7 places right)

  • Example 18 (Converting to Decimal Form):

  • a) $9.34 \times 10^5 = 934000$

  • b) $4.71 \times 10^{-5} = 0.0000471$

Scientific Notation using Significant Figures

• Rules for Counting Significant Figures:

  • Count digits from left to right starting at the first non-zero digit.

  • Zeros to the left of the first non-zero digit are not significant.

  • Examples:

  • $38041$ has 5 significant figures.

  • $0.0016$ has 2 significant figures.

  • $3.21 \times 10^4$ has 3 significant figures.

• Scientific Notation and Significant Figures:

  • In scientific notation $a \times 10^m$, exactly one non-zero significant figure sits to the left of the decimal point, and all digits shown in $a$ are significant.

• Palaeontology Application:

  • Estimated timeframe for Tyrannosaurus Rex on Earth: $7 \times 10^7$ to $6.5 \times 10^7$ years ago.

• Calculator Notation:

  • Calculators represent $2.37 \times 10^5$ as 2.37E5 or 2.37EE5 or using EXP / x10^x keys.

• Worked Examples:

  • Example 19 (Counting Significant Figures):

  • a) $451 \implies 3\text{ significant figures}$

  • b) $0.005012 \implies 4\text{ significant figures}$

  • c) $3.2 \times 10^7 \implies 2\text{ significant figures}$

  • Example 20 (Rounding to Significant Figures):

  • a) $2183000 = 2.18 \times 10^6$ (correct to 3 significant figures)

  • b) $0.0019482 = 1.95 \times 10^{-3}$ (correct to 3 significant figures)

  • Example 21 (Calculator Operations):

  • a) $(3.67 \times 10^5) \times (23.6 \times 10^4) = 8.6612 \times 10^{10} = 8.661 \times 10^{10}$ (to 4 sig. figs.)

  • b) $7.6 \times 10^{-3} + \sqrt{2.4 \times 10^{-2}} = 0.162518\dots = 1.625 \times 10^{-1}$ (to 4 sig. figs.)

Fractional Indices and Surds (Extending)

• Relationship Between Radical Notation and Fractional Indices:

  • A fractional power represents a root:

$a^{\frac{1}{m}} = \sqrt[m]{a}$

• Specifically:

$a^{\frac{1}{2}} = \sqrt{a} \quad (\text{square root}), \quad a^{\frac{1}{3}} = \sqrt[3]{a} \quad (\text{cube root})$

• General Fractional Exponent:

$a^{\frac{n}{m}} = (a^{\frac{1}{m}})^n = (\sqrt[m]{a})^n = \sqrt[m]{a^n}$

• Rational Numbers vs. Surds (Irrational Numbers):

  • Rational Numbers: Can be expressed exactly as a fraction $\frac{p}{q}$ ($q \neq 0$).

  • Example: $9^{\frac{1}{2}} = \sqrt{9} = 3 = \frac{3}{1}$

  • Surds: Irrational numbers written with a root sign whose decimal expansion is infinite and non-recurring with no repeating pattern.

  • Example: $5^{\frac{1}{3}} = \sqrt[3]{5} = 1.70997594668\dots$

  • Example: $\sqrt{2} = 1.41421356237\dots$

  • Example: $\sqrt{3} = 1.73205080757\dots$

• Classical Guitar Fret Spacing Application:

  • For a scale length $L\,\text{mm}$, the distance $x\,\text{mm}$ of the $n$-th fret from the bridge is modeled by:

$x = L \times 2^{-\frac{n}{12}}$

• At the 12th fret: $x = L \times 2^{-\frac{12}{12}} = L \times 2^{-1} = \frac{L}{2}$ (halving distance, doubling pitch to one octave higher).

  • Worked Examples:

• Example 22 (Writing with Root Signs):

  • a) $6^{\frac{1}{2}} = \sqrt{6}$

  • b) $2^{\frac{1}{5}} = \sqrt[5]{2}$

• Example 23 (Evaluating Fractional Powers):

  • a) $144^{\frac{1}{2}} = \sqrt{144} = 12$

  • b) $27^{\frac{1}{3}} = \sqrt[3]{27} = 3$

• Example 24 (Index Laws with Fractional Indices):

  • a) $a^{\frac{1}{2}} \times a^{\frac{3}{2}} = a^{\frac{1}{2} + \frac{3}{2}} = a^{\frac{4}{2}} = a^2$

  • b) $\frac{x^{\frac{1}{2}}}{x^{\frac{1}{3}}} = x^{\frac{1}{2} - \frac{1}{3}} = x^{\frac{3}{6} - \frac{2}{6}} = x^{\frac{1}{6}}$

  • c) $(y^2)^{\frac{1}{4}} = y^{2 \times \frac{1}{4}} = y^{\frac{2}{4}} = y^{\frac{1}{2}}$

Simple Operations with Surds (Extending)

• Addition and Subtraction of Surds:

  • Surds can only be added or subtracted if they are 'like' surds (contain the exact same number under the radical sign).

  • Examples:

$2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}$

$11\sqrt{7} - 2\sqrt{7} = 9\sqrt{7}$

$\sqrt{3} + \sqrt{5} \quad \text{(cannot be simplified, as they are unlike surds)}$

• Key Operations and Identities:

  • Square of a Radical:

$(\sqrt{a})^2 = a$

• Multiplication Rule for Surds:

$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$

• Division Rule for Surds:

$\sqrt{a} \div \sqrt{b} = \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

• Spacecraft Escape Velocity Application:

  • Formula for a spacecraft's escape velocity from gravity:

$v = \sqrt{2} \sqrt{\frac{GM}{R}}$

• Worked Examples:

  • Example 25 (Adding and Subtracting Surds):

  • a) $2\sqrt{5} + 6\sqrt{5} = 8\sqrt{5}$

  • b) $\sqrt{3} - 5\sqrt{3} = -4\sqrt{3}$

  • Example 26 (Multiplying and Dividing Surds):

  • a) $\sqrt{3} \times \sqrt{10} = \sqrt{3 \times 10} = \sqrt{30}$

  • b) $\sqrt{24} \div \sqrt{8} = \sqrt{\frac{24}{8}} = \sqrt{3}$

• Simplifying Surds using Perfect Square Factors:

  • If the number under the root sign has a factor that is a perfect square ($4, 9, 16, 25, 36, 49, 64, 81, 100$, etc.), extract it:

$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$

$\sqrt{8} - \sqrt{2} = \sqrt{4 \times 2} - \sqrt{2} = 2\sqrt{2} - \sqrt{2} = \sqrt{2}$

Applications, Technology, and Investigations

• Geometric Construction of Surds on a Number Line:

• Surds can be located precisely on a number line using right-angled triangle diagonals and circles (Pythagoras' Theorem: $c^2 = a^2 + b^2 \implies c = \sqrt{a^2 + b^2}$):

  • Unit square ($1 \times 1$): Diagonal length $c = \sqrt{1^2 + 1^2} = \sqrt{2} \approx 1.4142$

  • Rectangle $1 \times 2$: Diagonal length $c = \sqrt{1^2 + 2^2} = \sqrt{5} \approx 2.2361$

  • Constructing $\sqrt{6}$: Use a rectangle with sides $2$ and $\sqrt{2}$: Diagonal length $c = \sqrt{2^2 + (\sqrt{2})^2} = \sqrt{4 + 2} = \sqrt{6}$

  • Cell Division and Growth (Mitosis):

• Single cell dividing into 2 every hour: $N = 2^n$

• Single cell dividing into 3 every hour: $N = 3^n$

• Splitting every 20 minutes ($3$ cycles per hour) for 4 hours ($12$ cycles): $N = 2^{12} = 4096\,\text{cells}$

  • Physics Application (Universal Law of Gravitation):

• Gravitational force formula:

$F = \frac{G m_1 m_2}{d^2}, \quad \text{where } G = 6.67 \times 10^{-11}\,\text{N}\text{m}^2\text{kg}^{-2}$

• Gravitational acceleration formula $a = \frac{Gm}{r^2}$.

  • Surface acceleration on Earth ($m = 5.98 \times 10^{24}\,\text{kg}, r = 6.375 \times 10^6\,\text{m}$):

$a_{\text{Earth}} = \frac{(6.67 \times 10^{-11}) \times (5.98 \times 10^{24})}{(6.375 \times 10^6)^2} \approx 9.81\,\text{m/s}^2$

- Surface acceleration on Mars ($m = 6.42 \times 10^{23}\,\text{kg}, r = 3.37 \times 10^6\,\text{m}$):

$a_{\text{Mars}} = \frac{(6.67 \times 10^{-11}) \times (6.42 \times 10^{23})}{(3.37 \times 10^6)^2} \approx 3.77\,\text{m/s}^2$