Topic 1: Rounding and Approximations

Rounding and Approximations

• Rounding Targets and Categories

  • Numbers can be rounded according to different criteria depending on the degree of precision required:
    • Nearest Whole Number and Place Values: Rounding to units, tens, hundreds, thousands, and larger place-value units.
    • Decimal Places ($\text{d.p.}$): Rounding to a specified number of digits following the decimal point (e.g., $1\,\text{d.p.}$, $2\,\text{d.p.}$).
    • Significant Figures ($\text{s.f.}$): Rounding based on the most meaningful digits of a number, starting from the first non-zero digit.

• Fundamental Rounding Rules

  • Identify the deciding digit directly to the right of the specified rounding position:
    • Round Up: If the deciding digit is $5$ or greater ($\ge 5$), increase the target digit by $1$.
    • Round Down: If the deciding digit is $4$ or less ($\le 4$), leave the target digit unchanged and drop or replace subsequent digits with zeros.

• Standard Rounding Examples

  • Nearest $10$:
    • $4538 \approx 4540$ (the units digit is $8 \ge 5$, so round up the tens digit to $4$).
  • Nearest $100$:
    • $55,997 \approx 56,000$ (the tens digit is $9 \ge 5$, so the hundreds digit $9$ rounds up, causing a carry-over into the thousands column).
  • Correct to $1\,\text{d.p.}$:
    • $53.859 \approx 53.9$ (the second decimal place digit is $5 \ge 5$, so round up the first decimal place digit from $8$ to $9$).
  • Nearest Whole Number:
    • $20.09 \approx 20$ (the tenths digit is $0 \le 4$, so round down, keeping the integer value as $20$).

• Rounding Deduction Challenge

  • Clues Provided:
    • The value is strictly larger than $6$.
    • When rounded to the nearest whole number, it rounds down.
    • The ones digit is $0$.
    • The tenths digit is an even number.
  • Candidate Options:
    • $8.7$, $10.27$, $9.51$, $10.3$, $10.12$, $7.83$.
  • Step-by-Step Elimination:
    • $8.7$: Has ones digit $8$ (not $0$); rounds up to $9$.
    • $9.51$: Has ones digit $9$ (not $0$); rounds up to $10$.
    • $7.83$: Has ones digit $7$ (not $0$); rounds up to $8$.
    • $10.3$: Ones digit is $0$, rounds down to $10$, but tenths digit is $3$ (odd, not even).
    • $10.12$: Ones digit is $0$, rounds down to $10$, but tenths digit is $1$ (odd, not even).
    • $10.27$: Greater than $6$; rounds down to $10$; ones digit is $0$; tenths digit is $2$ (even).
  • Solution: $10.27$.

• Significant Figures ($\text{s.f.}$)

  • Core Identification Rule: The first significant figure is the first non-zero digit when reading from left to right. All zeros that appear after this first significant figure are considered significant.
  • Worked Examples:
    • $0.0123$ corrected to $2\,\text{s.f.} = 0.012$ (the first non-zero digit is $1$, followed by $2$; the next digit is $3 \le 4$, so round down).
    • $1.005$ corrected to $3\,\text{s.f.} = 1.01$ (the first non-zero digit is $1$, followed by significant zeros; the deciding fourth digit is $5 \ge 5$, so round up the third digit to $1$).
    • $6,895$ corrected to $3\,\text{s.f.} = 6,900$ (the deciding fourth digit is $5 \ge 5$, rounding up the $9$ to $10$, which shifts the hundred digit to $9$; note that trailing zeros in whole numbers indicate the place value).
    • Additional notation: $6.400$ represents $4\,\text{s.f.}$ precision.

• Rough Estimates

  • Guiding Principle: Unless stated otherwise, calculate a rough estimate by rounding each number in the expression to exactly $1\,\text{s.f.}$ before performing operations.
  • Square Number Benchmark:
    • The square number closest to $50$ is $49$ ($7^2 = 49$).
  • Worked Estimation Examples:
    • $8.74 \times 4.61 \approx 9 \times 5 = 45$
    • $9.55 - 5.82 \approx 10 - 6 = 4$
    • $2.9^2 + \sqrt{50} \approx 3^2 + \sqrt{49} = 9 + 7 = 16$
    • Calculation approximations: $9.71 \approx 10$, $23 + 28 \approx 20 + 30 = 50$, $\frac{2}{64} \approx 0.03125$, and multi-step evaluations such as $\frac{6(2+3) - 30}{10} = 0.5$.

Indices: Definitions, Operations, and Laws

• Index Notation and Terminology

  • Repeated Multiplication: Expressing repeated factors using a base and exponent, such as $7 \times 7 \times 7 \times 7 \times 7 = 7^5$.
  • Base and Index: In the expression $3^4$, the number $3$ is the base and $4$ is the index (or exponent/power).
  • Meaning of Index: The index indicates how many times the base number must be multiplied by itself.
  • Power of $1$ Property: Any number raised to the power of $1$ stays the same ($a^1 = a$); powers of $1$ do not need to be explicitly written.

• Sign Rules for Powers of Negative Numbers

  • Odd Exponents: If a negative number is raised to an odd power, the result is negative:
    • $(-1)^5 = -1$
  • Even Exponents: If a negative number is raised to an even power, the result is positive:
    • $(-1)^4 = 1$

• Direct Evaluation of Powers

  • $5^3 = 5 \times 5 \times 5 = 125$
  • $9^0 = 1$ (any number with an index of zero evaluates to $1$)
  • $6^3 = 6 \times 6 \times 6 = 216$

• Arithmetic Expressions Involving Powers

  • $3^2 \times 2^3 = (3 \times 3) \times (2 \times 2 \times 2) = 9 \times 8 = 72$
  • $3^4 - 2^4 = (3 \times 3 \times 3 \times 3) - (2 \times 2 \times 2 \times 2) = 81 - 16 = 65$
  • $8^2 + 3^3 - 2^4 = (8 \times 8) + (3 \times 3 \times 3) - (2 \times 2 \times 2 \times 2) = 64 + 27 - 16 = 75$
  • $9^2 \div 3^1 = (9 \times 9) \div 3 = 81 \div 3 = 27$

• Index Law 1: Multiplying Numbers in Index Form (Product Rule)

  • Rule: To multiply numbers having the same base, add the indices together, including any negative indices:
    • $a^m \times a^n = a^{m+n}$
  • Condition: Both base numbers must be strictly identical (e.g., $2^3 \times 2^2 = 2^5$, but $2^3 \times 5^4$ cannot be combined by adding powers).
  • Examples:
    • $2^3 \times 2^{-2} = 2^{3 + (-2)} = 2^{3 - 2} = 2^1 = 2$
    • $11^5 \times 11^3 \times 11^{-2} = 11^{5 + 3 - 2} = 11^6$
    • $3^1 \times 3^1 \times 3^2 \times 3^4 = 3^{1 + 1 + 2 + 4} = 3^8$

• Index Law 2: Dividing Numbers in Index Form (Quotient Rule)

  • Rule: To divide numbers having the same base, subtract the denominator index from the numerator index, including negative indices:
    • $\frac{a^m}{a^n} = a^{m-n}$
  • Expansion Verification:
    • $\frac{2^2}{2^4} = \frac{2 \times 2}{2 \times 2 \times 2 \times 2} = \frac{1}{2 \times 2} = \frac{1}{2^2} = 2^{-2}$
    • Direct index subtraction: $2^{2 - 4} = 2^{-2}$
  • Examples:
    • $\frac{3^5}{3^3} = 3^{5 - 3} = 3^2$
    • $4^2 \div 4 \times 4^2 = 4^2 \div 4^1 \times 4^2 = 4^{2 - 1 + 2} = 4^3$

• Index Law 3: The Zero Index

  • Rule: Any non-zero number raised to the power of $0$ is equal to $1$:
    • $a^0 = 1$
  • Examples:
    • $5^0 = 1$
    • $23^0 = 1$
    • $43^0 = 1$

• Index Law 4: Negative Indices and Reciprocals

  • Rule: Any number with a negative index is equal to its reciprocal with a positive index:
    • $a^{-1} = \frac{1}{a}$
    • $a^{-m} = \frac{1}{a^m}$
    • $\left(\frac{a}{b}\right)^{-m} = \left(\frac{b}{a}\right)^m = \frac{b^m}{a^m}$
  • Worked Examples:
    • $3^{-4} = \frac{1}{3^4} = \frac{1}{81}$
    • $\left(\frac{2}{3}\right)^{-4} = \left(\frac{3}{2}\right)^4 = \frac{3^4}{2^4} = \frac{81}{16} = 5\frac{1}{16}$ (converted from improper fraction to mixed number)
    • $7^{-4} = \frac{1}{7^4}$
    • $20^{-1} = \frac{1}{20^1} = \frac{1}{20}$
  • Arithmetic Computations with Negative Indices:
    • $2^{-3} + 4^{-2} = \frac{1}{2^3} + \frac{1}{4^2} = \frac{1}{8} + \frac{1}{16} = \frac{2}{16} + \frac{1}{16} = \frac{3}{16}$
    • $5^2 - 2^{-1} = 25 - \frac{1}{2} = 24\frac{1}{2}$
    • $4^{-2} \times 3^2 = \frac{1}{4^2} \times 9 = \frac{1}{16} \times 9 = \frac{9}{16}$

• Index Law 5: Power of a Power

  • Rule: To raise a power to another power, multiply the indices together:
    • $(a^m)^n = a^{m \times n} = a^{mn}$
  • Examples:
    • $(4^3)^{-4} = 4^{3 \times (-4)} = 4^{-12}$
    • $(5^3)^0 = 5^{3 \times 0} = 5^0 = 1$
    • $(7^4)^5 = 7^{4 \times 5} = 7^{20}$
    • $(3x^3)^4 = 3^4 \times (x^3)^4 = 81x^{12}$
    • $\frac{(4p^5)^2}{2p} = \frac{16p^{10}}{2p} = 8p^{10 - 1} = 8p^9$
    • $(ka^j)^2 = k^2 a^{2j}$

• Summary of Index Laws

  • Multiplication: $a^m \times a^n = a^{m+n}$
  • Division: $\frac{a^m}{a^n} = a^{m-n}$
  • Zero Power: $a^0 = 1$
  • Negative Exponent: $a^{-m} = \frac{1}{a^m}$
  • Power of a Power: $(a^m)^n = a^{mn}$

• Ordering Values from Smallest to Greatest (Ascending Order)

  • Terminology Note: Ascending indicates increasing order ($\uparrow$); descending indicates decreasing order ($\downarrow$).
  • Ordering Set 1:
    • Given: $3^4$, $\sqrt{81}$, $10^2$, $43^0$, $2\sqrt{49}$
    • Evaluate each term: $3^4 = 81$, $\sqrt{81} = 9$, $10^2 = 100$, $43^0 = 1$, $2\sqrt{49} = 2 \times 7 = 14$
    • Ascending order: $43^0, \sqrt{81}, 2\sqrt{49}, 3^4, 10^2$ ($1 < 9 < 14 < 81 < 100$)
  • Ordering Set 2:
    • Given: $5^{-2}$, $9^1$, $6^{-1}$, $3^{-2}$, $4^0$
    • Evaluate each term: $5^{-2} = \frac{1}{25} = 0.04$, $9^1 = 9$, $6^{-1} = \frac{1}{6} \approx 0.167$, $3^{-2} = \frac{1}{9} \approx 0.111$, $4^0 = 1$
    • Exponent hint: Powers of $4$ producing $64$ include $4^3 = 64$ and $4^{-3} = \frac{1}{64}$
    • Ascending order: $5^{-2}, 3^{-2}, 6^{-1}, 4^0, 9^1$ ($\frac{1}{25} < \frac{1}{9} < \frac{1}{6} < 1 < 9$)

• Solving for Unknown Exponents and Bases

  • $10^x = 100,000 \implies 10^x = 10^5 \implies x = 5$
  • $x^{-3} = \frac{1}{64} \implies \frac{1}{x^3} = \frac{1}{4^3} \implies x = 4$
  • $5^0 = x \implies 1 = x \implies x = 1$

• Standard Form (Scientific Notation)

  • Definition and Method: Shift the decimal point so that it is positioned directly after the first non-zero digit.
  • Exponent Sign Conventions:
    • Shift decimal point to the left: power of $10$ is positive (\times 10^{+n}$).\n * Shift decimal point to the **right**: power of 10$is negative ($\times 10^{-n}$).
  • Example:
    • $36,400 = 3.64 \times 10^4$

• Mixed Index Exercises

  • $3^2 + 4^3 - 5^0 = 9 + 64 - 1 = 72$
  • $\frac{k^3 \times k^5}{k^2} = \frac{k^8}{k^2} = k^{8 - 2} = k^6$
  • $a^3 \times a^0 \times a^{-5} = a^{3 + 0 - 5} = a^{-2}$
  • $(n^{-5})^{-2} = n^{(-5) \times (-2)} = n^{10}$
  • $19^8 \div 19^{11} = 19^{8 - 11} = 19^{-3} = \frac{1}{19^3}$ (expressed in positive index form)
  • $\frac{2^3 \times 4^{-2}}{8} = \frac{8 \times \frac{1}{16}}{8} = \frac{\frac{1}{2}}{8} = \frac{1}{16}$ (expressed as a simplified fraction)
  • $\frac{27a^2b^4}{3a^2b} = \frac{27}{3} \times \frac{a^2}{a^2} \times \frac{b^4}{b^1} = 9 \times 1 \times b^{4 - 1} = 9b^3$
  • $\frac{9^{-3} \times 9^1 \times 9^{-2}}{9^{-4}} = \frac{9^{-3 + 1 - 2}}{9^{-4}} = \frac{9^{-4}}{9^{-4}} = 9^{-4 - (-4)} = 9^0 = 1$
  • $\frac{a^3 \times a^2}{a^4} = \frac{a^{3 + 2}}{a^4} = \frac{a^5}{a^4} = a^{5 - 4} = a^1 = a$

Algebraic Expressions: Simplification and Single Brackets

• Classification of Expressions

  • Binomial: An algebraic expression consisting of exactly two terms (e.g., $a + b$).
  • Trinomial: An algebraic expression consisting of exactly three terms (e.g., $a + b + c$).
  • Polynomial: An algebraic expression consisting of multiple terms (e.g., $a + b + c + d + \dots$).

• Collecting Like Terms

  • Fundamental Principle: Only terms containing the exact same variables raised to the exact same powers can be combined.
  • Critical Algebraic Caveat: $x^2$ and $x$ are not the same; they represent different powers and cannot be added or combined into a single term.
  • Signed Operations on Linear Variables:
    • $-x - x = -2x$
    • $-x + 5x - x = 3x$
    • $5x - 8x + 7x = 4x$
  • Handling Equivalent Variable Order (Commutativity):
    • Because $pq = qp$ and $rs = sr$, like terms can be combined across order variations:
    • $15pq - 7rs + 8qp + 2sr = (15 + 8)pq + (-7 + 2)rs = 23pq - 5rs$
  • Worked Simplification Examples:
    • $-x + 3x^2 + 5x - x = -x + 5x - x + 3x^2 = 3x + 3x^2$
    • $4x + 7y - x - 5y = (4 - 1)x + (7 - 5)y = 3x + 2y$
    • $8x^2 + 4x + 3 - 3x^2 + 7x - 8 = (8 - 3)x^2 + (4 + 7)x + (3 - 8) = 5x^2 + 11x - 5$
    • $m + m - m - n = m - n$
    • $5wx + 3x^2 - 6wx + 7wx - x^2 = (5 - 6 + 7)wx + (3 - 1)x^2 = 6wx + 2x^2$

• Expanding Single Brackets

  • Distributive Law: Multiply the term outside the bracket by every term inside the bracket.
  • Sign Reversal Rule: When multiplying by a negative number outside the bracket, reverse the signs of all terms inside the bracket.
  • Basic Single Bracket Expansions:
    • $2(x + 4) = 2x + 8$
    • $-3(1 - a) = -3 + 3a$
    • $2(5 - 2y + 3w) = 10 - 4y + 6w$
    • $5n(12 - 3n) = 60n - 15n^2$
    • $3g(2g^2 + 7) = 6g^3 + 21g$
    • $y(3y^2 - 4y + 5) = 3y^3 - 4y^2 + 5y$

• Expanding and Simplifying Multi-Bracket Linear Expressions

  • $11(x + 4) + 3(x - 7) = 11x + 44 + 3x - 21 = (11 + 3)x + (44 - 21) = 14x + 23$
  • $7(10t - 2) - 2(9 + 3t) = 70t - 14 - 18 - 6t = (70 - 6)t + (-14 - 18) = 64t - 32$
  • $5x + 2(3x + 1) = 5x + 6x + 2 = 11x + 2$
  • $3x + 7 - 2(2x + 5) = 3x + 7 - 4x - 10 = (3 - 4)x + (7 - 10) = -x - 3$

Quadratic Expansion, Monomial Operations, and Factorisation

• Expanding Two Brackets (Binomials)

  • Quadratic Expression Definition: An algebraic expression where the highest exponent of the variable is $2$ (squared), such as $x^2 + 8x + 15$.
  • Sign Rules for Double Bracket Products:
    • $(? - ?)(? - ?) \rightarrow$ constant term is positive ($(-)\times(-) = (+)$).
    • $(? + ?)(? - ?) \rightarrow$ constant term is negative ($(+)\times(-) = (-)$).
    • $(? + ?)(? + ?) \rightarrow$ constant term is positive ($(+)\times(+) = (+)$).
  • Worked Double Bracket Expansions:
    • $(x - 2)(x + 5) = x^2 + 5x - 2x - 10 = x^2 + 3x - 10$
    • $(m + 4)(m - 5) = m^2 - 5m + 4m - 20 = m^2 - m - 20$

• Perfect Squares

  • Definition: A perfect square expression occurs when a binomial bracket is multiplied directly by itself (e.g., $(x + 8)^2 = (x + 8)(x + 8)$).
  • Worked Perfect Square Expansions:
    • $(x - 3)^2 = (x - 3)(x - 3) = x^2 - 3x - 3x + 9 = x^2 - 6x + 9$
    • $(y - 30)^2 = (y - 30)(y - 30) = y^2 - 30y - 30y + 900 = y^2 - 60y + 900$

• Multiplying Algebraic Monomials

  • Multiply numerical coefficients together and apply index laws to multiply matching variable bases:
    • $2x^2 \times 5y = 10x^2y$
    • $4m^2n^2 \times 7mn^2 = (4 \times 7)(m^2 \times m^1)(n^2 \times n^2) = 28m^3n^4$
    • $3cd^3 \times 4c = (3 \times 4)(c^1 \times c^1)(d^3) = 12c^2d^3$
    • $y^2 \times 8y = 8(y^2 \times y^1) = 8y^3$
    • $5z^1 \times 2z^3 \times 3z = (5 \times 2 \times 3)(z^1 \times z^3 \times z^1) = 30z^{1 + 3 + 1} = 30z^5$

• Cancelling and Simplifying Algebraic Fractions

  • Divide coefficients by their highest common numerical factor and subtract variable powers using the quotient rule:
    • $\frac{10ab^2c}{25abc} = \frac{10}{25} \times \frac{a}{a} \times \frac{b^2}{b} \times \frac{c}{c} = \frac{2}{5}b = 0.4b$
    • $\frac{30x^3}{12xy} = \frac{30}{12} \times \frac{x^3}{x} \times \frac{1}{y} = \frac{5x^2}{2y}$
    • $\frac{48x^6}{30x^4} = \frac{48 \div 6}{30 \div 6} x^{6 - 4} = \frac{8}{5}x^2$

• Factorising Expressions (Extracting Common Factors)

  • Definition: The reverse process of expanding brackets; transforming an expression into a product of its highest common factor (HCF) and a bracketed expression.
  • Single-Variable and Multi-Variable Factorisation Examples:
    • $12pq + 28qr = 4q(3p + 7r)$ (HCF is $4q$)
    • $3y^2 - 9y = 3y(y - 3)$ (HCF is $3y$)
    • $4x + x^2y = x(4 + xy)$ (HCF is $x$)
    • $2x^2 + 8y^2 - 4 = 2(x^2 + 4y^2 - 2)$ (HCF is $2$)
    • $3t^2 - 6t = 3t(t - 2)$ (HCF is $3t$)
    • $6a^2b + 9ab^3 = 3ab(2a + 3b^2)$ (HCF is $3ab$)
    • $5ef + 6e^2f = ef(5 + 6e)$ (HCF is $ef$)
    • $6gh^2 + 6gh - 3g^2h = 3gh(2h + 2 - g)$ (HCF is $3gh$)