MATHEMATICS HIP-HOP ACADEMY · CHAPTER 01
Numerical Concepts: Surds, Standard Form and Numerical Approximations
A-Level Pure Mathematics · Part 3 · Teaching notes, worked examples and practice exercises
About this lesson
This chapter continues Numerical Concepts in the Mathshub Uganda A-Level Principal Mathematics series. It covers surds, standard form, and numerical approximations. The core explanations and exercises are suitable for Senior Five learning and Senior Six revision. Teachers should cross-check the current school scheme of work and NCDC syllabus document when assigning coverage.
Learning objectives: by the end of this lesson, learners should be able to simplify surds; perform operations with surds; rationalise simple denominators; express very large or small numbers in standard form; convert between standard form and ordinary notation; and round numbers to a stated degree of accuracy.
1. Surds
A surd is an irrational root expression that cannot be simplified to a rational number. For example, √2 and √7 are surds. However, √25 = 5 is not a surd because it is rational.
1.1 Simplifying surds
Look for a perfect-square factor inside the square root. Use √(ab) = √a × √b for non-negative a and b.
Worked Example 1
Simplify √72.
√72 = √(36 × 2) = √36 × √2 = 6√2.
Worked Example 2
Simplify √180.
√180 = √(36 × 5) = 6√5.
Worked Example 3
Simplify 3√12 − √27.
3√12 − √27 = 3(2√3) − 3√3 = 6√3 − 3√3 = 3√3.
1.2 Adding and subtracting surds
Only like surds can be combined. Simplify first, then collect terms with the same root.
Example: 2√8 + √18 = 2(2√2) + 3√2 = 4√2 + 3√2 = 7√2.
But √2 + √3 cannot be combined into a single like-surd term.
1.3 Multiplying surds
Use √a × √b = √(ab) for non-negative a and b, or multiply the coefficients and the surd parts separately.
Example: √6 × √15 = √90 = 3√10.
Example: (2√3)(5√6) = 10√18 = 30√2.
1.4 Rationalising a denominator
Rationalising removes a surd from the denominator of a fraction.
Worked Example 4
Simplify 5/√3 by rationalising the denominator.
Multiply numerator and denominator by √3: (5/√3) × (√3/√3) = 5√3/3.
Worked Example 5
Simplify 4/(2 + √3).
Multiply by the conjugate, 2 − √3: [4/(2 + √3)] × [(2 − √3)/(2 − √3)] = 4(2 − √3)/(4 − 3) = 8 − 4√3.
The conjugate of a + √b is a − √b. Their product is a² − b, which can remove the surd from the denominator when a² − b is non-zero.
2. Standard form
1.5 Fractional surds
A fractional surd is a fraction that contains a surd in its numerator, denominator, or both. Some fractions involving surds are already simplified; others can be simplified or rationalised. Keep exact surd answers unless a decimal approximation is requested.
Worked Example 6: Simplifying a fractional surd
Simplify √12/√3.
√12/√3 = √(12/3) = √4 = 2.
Worked Example 7: Simplifying a fraction with a surd numerator
Simplify (3√8)/4.
(3√8)/4 = (3 × 2√2)/4 = 6√2/4 = 3√2/2.
Worked Example 8: Rationalising a fractional surd
Simplify 2/√5.
Multiply top and bottom by √5: (2/√5) × (√5/√5) = 2√5/5.
Worked Example 9: Rationalising with a conjugate
Simplify 3/(2 + √2).
Multiply top and bottom by 2 − √2: 3/(2 + √2) × (2 − √2)/(2 − √2) = 3(2 − √2)/(4 − 2) = 3(2 − √2)/2.
Worked Example 10: Adding fractional surds
Simplify √3/2 + √3/4.
Use the common denominator 4: √3/2 + √3/4 = 2√3/4 + √3/4 = 3√3/4.
Important: only like surds can be combined. For example, √2/3 + √3/3 cannot be combined into one like-surd term.
Standard form (scientific notation) writes a number as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.
Worked Example 6: Large number
Write 5,830,000 in standard form.
Move the decimal point six places left: 5,830,000 = 5.83 × 10⁶.
Worked Example 7: Small number
Write 0.0000724 in standard form.
Move the decimal point five places right to obtain 7.24, so 0.0000724 = 7.24 × 10⁻⁵.
Worked Example 8: Convert to ordinary notation
Write 3.06 × 10⁴ and 8.2 × 10⁻³ as ordinary numbers.
3.06 × 10⁴ = 30,600; 8.2 × 10⁻³ = 0.0082.
2.1 Operations in standard form
Multiply: multiply the decimal parts and add the powers of ten. Divide: divide the decimal parts and subtract the powers of ten. Finally, rewrite the answer so its first factor is at least 1 and less than 10.
Worked Example 9
Calculate (3 × 10⁵)(4 × 10⁻²).
(3 × 4) × 10⁽⁵⁻²⁾ = 12 × 10³ = 1.2 × 10⁴.
Worked Example 10
Calculate (8.4 × 10⁶)/(2.1 × 10²).
(8.4/2.1) × 10⁽⁶⁻²⁾ = 4 × 10⁴.
3. Numerical approximations
A numerical approximation is a value close to the exact value. Always follow the requested degree of accuracy: decimal places (d.p.), significant figures (s.f.), or a stated tolerance.
3.1 Decimal places
Decimal places count digits after the decimal point. To round to a given number of decimal places, inspect the next digit: 0–4 means leave the last retained digit unchanged; 5–9 means increase it by one.
Example: 8.3764 to 2 decimal places is 8.38, because the third decimal digit is 6.
3.2 Significant figures
Significant figures begin at the first non-zero digit. Zeros between non-zero digits count; leading zeros do not count.
Example: 0.004786 to 3 significant figures is 0.00479.
Example: 53,684 to 2 significant figures is 54,000 (or 5.4 × 10⁴ to show the significant figures clearly).
3.3 Upper and lower bounds (extension)
When a measurement is rounded, the true value lies within a range. If a length is given as 12.4 cm correct to the nearest 0.1 cm, its lower bound is 12.35 cm and its upper bound is 12.45 cm, usually written 12.35 ≤ L < 12.45.
Worked Example 11
A mass is 6.8 kg correct to the nearest 0.1 kg. State its lower and upper bounds.
Half of 0.1 is 0.05. Therefore 6.75 ≤ m < 6.85 kg.
4. Mixed worked examples
Worked Example 12
Simplify (√50 + √8) / √2.
√50 = 5√2 and √8 = 2√2. Thus (5√2 + 2√2)/√2 = 7√2/√2 = 7.
Worked Example 13
A bacterium has a mass of 0.0000000046 g. Express this in standard form.
0.0000000046 = 4.6 × 10⁻⁹ g.
Worked Example 14
Round 0.078946 to 3 significant figures.
The first three significant digits are 7, 8 and 9. The next digit is 4, so the result is 0.0789.
5. Practice exercises
Attempt all questions before checking the answers. Give exact answers in surd questions unless an approximation is requested.
1. Simplify √48.
2. Simplify √200.
3. Simplify 4√18 − √50.
4. Simplify √12 × √15.
5. Expand and simplify (√5 + 2)(√5 − 2).
6. Rationalise the denominator of 7/√5.
7. Simplify 3/(√5 − 2).
8. Write 6,720,000 in standard form.
9. Write 0.000000391 in standard form.
10. Write 4.08 × 10⁵ as an ordinary number.
11. Write 7.3 × 10⁻⁴ as an ordinary number.
12. Calculate (2.5 × 10⁴)(6 × 10⁻³), giving your answer in standard form.
13. Calculate (9.6 × 10⁷)/(3.2 × 10³), giving your answer in standard form.
14. Round 37.5862 to 2 decimal places.
15. Round 0.006784 to 2 significant figures.
16. Round 987,650 to 3 significant figures.
17. A length is 8.36 cm correct to the nearest 0.01 cm. State its lower and upper bounds.
18. Simplify (√75 − √12) / √3.
19. Rationalise and simplify 2/(3 + √5).
20. A number is 0.000583 correct to 3 significant figures. State the interval within which the original value lies.
6. Answers
1. √48 = √(16 × 3) = 4√3.
2. √200 = √(100 × 2) = 10√2.
3. 4√18 − √50 = 12√2 − 5√2 = 7√2.
4. √12 × √15 = √180 = 6√5.
5. (√5 + 2)(√5 − 2) = 5 − 4 = 1.
6. 7/√5 = 7√5/5.
7. 3/(√5 − 2) × (√5 + 2)/(√5 + 2) = 3(√5 + 2)/(5 − 4) = 3√5 + 6.
8. 6.72 × 10⁶.
9. 3.91 × 10⁻⁷.
10. 408,000.
11. 0.00073.
12. 15 × 10¹ = 1.5 × 10².
13. 3 × 10⁴.
14. 37.59.
15. 0.0068.
16. 988,000 (or 9.88 × 10⁵).
17. 8.355 ≤ L < 8.365 cm.
18. (5√3 − 2√3)/√3 = 3.
19. 2/(3 + √5) × (3 − √5)/(3 − √5) = (6 − 2√5)/(9 − 5) = (3 − √5)/2.
20. To 3 significant figures, the rounding unit is 0.000001; half-unit is 0.0000005. Thus 0.0005825 ≤ x < 0.0005835.
Fractional surds: extra practice
21. Simplify √18/√2.
22. Simplify (5√12)/6.
23. Rationalise the denominator of 4/√7.
24. Simplify 5/(3 − √2) by rationalising the denominator.
25. Simplify √5/3 + √5/6.
Answers to fractional surds practice
21. √18/√2 = √9 = 3.
22. (5√12)/6 = (5 × 2√3)/6 = 5√3/3.
23. 4/√7 = 4√7/7.
24. 5/(3 − √2) × (3 + √2)/(3 + √2) = 5(3 + √2)/(9 − 2) = (15 + 5√2)/7.
25. √5/3 + √5/6 = 2√5/6 + √5/6 = 3√5/6 = √5/2.
7. Summary checklist
☐ I can simplify surds by extracting perfect-square factors.
☐ I can add, subtract and multiply surds.
☐ I can rationalise simple denominators, including using a conjugate.
☐ I can convert numbers to and from standard form.
☐ I can multiply and divide numbers in standard form.
☐ I can round to decimal places and significant figures.
☐ I can state bounds for rounded measurements.
Teacher note: upper bounds are normally excluded when using a half-open interval for rounding. Check the expected notation and depth against the current school scheme of work.
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