006-MWALA_LEARN-SOLVING-FOREVER-2

Objectives: MATHEMATICS SOLVING EXPONENT-RADICALS

Exponential Questions

1. Give the base and exponent for each of the following exponential numbers:

  1. 617
  2. 74
  3. (-10)19
  4. 3100
  5. 62
  6. 50
  7. j25
  8. 176
  9. (1/6)9
  10. (1/2)17
  11. 19101
  12. (-75)8
  13. (x + y)n
  14. (7 + x)8

2. Express each of the following exponential numbers in expanded form:

  1. 72
  2. (-3)3
  3. 103
  4. 26
  5. 91
  6. (-y)3
  7. (-99)5
  8. (9/10)5
  9. (-0.35)4
  10. (0.67)3

3. Write each of the following expressions in exponential form and then give the base and the exponent:

  1. 7 Γ— 7 Γ— 7 Γ— 7
  2. (-2) Γ— (-2) Γ— (-2) Γ— (-2) Γ— (-2) Γ— (-2) Γ— (-2)
  3. 14 Γ— 14 Γ— 14
  4. 19
  5. (x + b)(x + b)(x + b)(x + b)
  6. (-r)(-r)(-r)(-r)(-r)(-r)(-r)
  7. 50 Γ— 50 Γ— ... Γ— 50 (30 times)
  8. 5 Γ— 5 Γ— 5 Γ— 5 Γ— 5 Γ— 5 Γ— 5 Γ— 5
  9. (a + b)(a + b)(a + b)(a + b)(a + b)
  10. (w Γ— w Γ— w Γ— w Γ— w Γ— w Γ— w Γ— w Γ— w Γ— w)
  11. 0.3 Γ— 0.3 Γ— 0.3 Γ— 0.3 Γ— 0.3
  12. v Γ— v Γ— v
Exponential Questions

4. Find the value of each of the following exponential numbers:

  1. 52
  2. 72
  3. 203
  4. 122
  5. 13
  6. (-9)2
  7. 303
  8. (-2)3
  9. (-3)3
  10. 25
  11. (-5)3
  12. 103

5. Express each of the following numbers in exponential form using the given bases:

  1. 25 in base 5
  2. 36 in base 6
  3. 1,728 in base 12
  4. 16 in base 2
  5. 1,000,000,000,000 in base 10

6. Express each of the following numbers as powers of two different bases:

  1. 16
  2. 64
  3. 81
  4. 625
  5. 1,000
Exponents β€” Full Notes, Methods, and Answers

Exponents β€” Full Notes, Multiple Methods & Worked Answers

β€œBase” is the number being multiplied repeatedly. β€œExponent (power)” shows how many times.

Quick Notes (Read this first!)

  • an means multiply a by itself n times. Example: 34 = 3Γ—3Γ—3Γ—3 = 81.
  • Even vs Odd powers with negatives: (-a)even is positive; (-a)odd is negative.
  • Special powers: a1=a; a0=1 (as long as aβ‰ 0).
  • Place-value shortcut: 10n is β€œ1 followed by n zeros”.
  • Prime-factor view: break numbers (like 20=22Γ—5). Then apply powers to each factor.
Real-life: Gaming XP or social followers doubling β†’ exponents. Phone storage β€œGB” grows by powers of 2. Compound interest uses powers (money β€œmultiplies itself”). Volumes of cubes use powers of 3.

1) Give the base and exponent

  1. 617 β†’ Base = 6, Exponent = 17
  2. 74 β†’ Base = 7, Exponent = 4
  3. (βˆ’10)19 β†’ Base = βˆ’10, Exponent = 19
  4. 3100 β†’ Base = 3, Exponent = 100
  5. 62 β†’ Base = 6, Exponent = 2
  6. 50 β†’ Base = 5, Exponent = 0 value = 1
  7. j25 β†’ Base = j, Exponent = 25
  8. 176 β†’ Base = 17, Exponent = 6
  9. (1/6)9 β†’ Base = 1/6, Exponent = 9
  10. (1/2)17 β†’ Base = 1/2, Exponent = 17
  11. 19101 β†’ Base = 19, Exponent = 101
  12. (βˆ’75)8 β†’ Base = βˆ’75, Exponent = 8
  13. (x + y)n β†’ Base = (x + y), Exponent = n
  14. (7 + x)8 β†’ Base = (7 + x), Exponent = 8

2) Expanded form (and values)

  1. 72 β†’ 7Γ—7 = 49
  2. (βˆ’3)3 β†’ (βˆ’3)Γ—(βˆ’3)Γ—(βˆ’3) = βˆ’27
    Odd power keeps the negative.
  3. 103 β†’ 10Γ—10Γ—10 = 1000
  4. 26 β†’ 2Γ—2Γ—2Γ—2Γ—2Γ—2 = 64
    Method 2 (doubling idea)1β†’2β†’4β†’8β†’16β†’32β†’64 (six doublings).
  5. 91 β†’ 9
  6. (βˆ’y)3 β†’ (βˆ’y)(βˆ’y)(βˆ’y) = βˆ’y3
  7. (βˆ’99)5 β†’ (βˆ’99)Γ—(βˆ’99)Γ—(βˆ’99)Γ—(βˆ’99)Γ—(βˆ’99) = negative
  8. (9/10)5 β†’ (9/10)Γ—(9/10)Γ—(9/10)Γ—(9/10)Γ—(9/10) = 95/105
  9. (βˆ’0.35)4 β†’ (βˆ’0.35)Γ—(βˆ’0.35)Γ—(βˆ’0.35)Γ—(βˆ’0.35) = positive
    Even power β†’ sign becomes +.
  10. (0.67)3 β†’ 0.67Γ—0.67Γ—0.67 = 0.300763 (approx.)

3) Write in exponential form (give base & exponent)

  1. 7Γ—7Γ—7Γ—7 β†’ 74 (base 7, exponent 4)
  2. (βˆ’2) repeated 7 times β†’ (βˆ’2)7 (base βˆ’2, exponent 7)
  3. 14Γ—14Γ—14 β†’ 143
  4. 19 β†’ 191
  5. (x+b)Γ—(x+b)Γ—(x+b)Γ—(x+b) β†’ (x+b)4
  6. (βˆ’r)Γ—(βˆ’r)Γ—(βˆ’r)Γ—(βˆ’r)Γ—(βˆ’r)Γ—(βˆ’r)Γ—(βˆ’r) β†’ (βˆ’r)7
  7. 50 Γ— 50 Γ— … Γ— 50 (30 times) β†’ 5030
  8. 5 multiplied 8 times β†’ 58
  9. (a+b) five times β†’ (a+b)5
  10. w multiplied 10 times β†’ w10
  11. 0.3 multiplied 5 times β†’ (0.3)5
  12. vΓ—vΓ—v β†’ v3

4) Find the value

  1. 52 β†’ 25
  2. 72 β†’ 49
  3. 203 β†’ 8000
    Method 2 (break 20)20=2Γ—10 β‡’ 203=(2Γ—10)3=23Γ—103=8Γ—1000=8000.
  4. 122 β†’ 144
  5. 13 β†’ 1
  6. (βˆ’9)2 β†’ 81
    Even power β†’ positive.
  1. 303 β†’ 27,000
    Why quickly?30=3Γ—10 β‡’ 33Γ—103=27Γ—1000=27,000.
  2. (βˆ’2)3 β†’ βˆ’8
  3. (βˆ’3)3 β†’ βˆ’27
  4. 25 β†’ 32
  5. (βˆ’5)3 β†’ βˆ’125
  6. 103 β†’ 1000
Real-life: A 30 cm cube has volume 303=27,000 cmΒ³. That’s exactly what we computed.

5) Write each number in exponential form using the given base

  1. 25 in base 5 β†’ 52
  2. 36 in base 6 β†’ 62
  3. 1,728 in base 12 β†’ 123
    Check by factors12=3Γ—4=3Γ—22. Then 123=33Γ—26=27Γ—64=1728.
  4. 16 in base 2 β†’ 24 2Γ—2Γ—2Γ—2
  5. 1,000,000,000,000 in base 10 β†’ 1012 12 zeros
Money example: Tsh 10,000 = 104. A trillion shillings is 1012.

6) Express each number as powers of two different bases

  1. 16 β†’ 24 and 42
  2. 64 β†’ 26 and 43 (also 82)
  3. 81 β†’ 34 and 92
  4. 625 β†’ 54 and 252
  5. 1,000 β†’ 103 and (1/10)βˆ’3 (another base, inverse)
Method (prime factors) 16=24 β‡’ (4)2. 64=26 β‡’ (4)3 β‡’ (8)2. 81=34 β‡’ (9)2. 625=54 β‡’ (25)2. 1000=103= (1/10)βˆ’3.
Gaming example: If a sword’s damage doubles every level: L5 damage = baseΓ—24. Same idea as 16=24.

Why these methods work (short summary)

  • Repeated multiplication model: Exponents compress long products; this is the definition.
  • Parity rule for negatives: Pairs of negatives make positives. Even powers are many pairs; odd powers leave one negative.
  • Base-10 shortcuts: Every power of 10 shifts digits β†’ fast mental math for thousands, millions, billions.
  • Factor-and-power: Writing numbers as products (e.g., 30=3Γ—10) lets you apply powers to each part separately: (ab)n=anbn.
  • Change of base idea: If a=bk, then am=(bk)m=bkm. That’s why 16=42 and also 24.
Everyday intuition: β€œSquared” = area of a square (side2). β€œCubed” = volume of a cube (edge3). That’s why powers grow fastβ€”adding one to the exponent jumps dimensions of multiplication.
Β© Learning helper β€” built for clarity. Soma maelezo mafupi (Notes) kwanza, kisha pitia majibu hatua kwa hatua.
Exponents β€” Full Step-by-Step Solutions

Exponents β€” Competence-Based Questions with Full Solutions

Hapa kila swali limeelezewa hatua kwa hatua, maelezo ya makosa ya kawaida, na mifano halisi ya maisha.

1) Write as a single power: 2 Γ— 2 Γ— 2 Γ— 2 Γ— 2
Repeated Multiplication β†’ Exponential Form

Solution:

  1. Count how many 2’s: 5 times
  2. Write as 2⁡

Answer: 25

Explanation: Exponent = how many times base multiplies itself. Mistake: writing 2Γ—5 (incorrect).

Example: 5 trees planted in a row, each branch splits into 2 β†’ total branches = 2⁡ = 32.

2) Evaluate exactly: (-3)4
Even Exponent & Negative Numbers

Solution:

  1. Even exponent = 4 β†’ negative raised to even power becomes positive.
  2. (-3)Β² = 9
  3. 9Β² = 81

Answer: 81

Example: Energy calculation: negative charge squared twice β†’ positive energy = 81 units.

3) Evaluate: -34 (no brackets)
Order of Operations & Notation

Solution:

  1. No brackets β†’ exponent applies only to 3
  2. Compute 3⁴ = 81
  3. Apply leading negative: -81

Answer: -81

Example: Debt: owing 3 dollars per day raised to 4th power β†’ total debt = -81 dollars.

4) Express 1,000,000 as a power of 10
Place Value Powers

Solution:

  1. Count zeros: 1,000,000 β†’ 6 zeros
  2. Write as 10⁢

Answer: 106

Example: 1 MB in computers β‰ˆ 10⁢ bytes.

5) Rewrite as a single fraction power: (3/5)Β³
Fraction Powers

Solution:

  1. (3/5)Β³ = 3Β³ / 5Β³
  2. 3Β³ = 27, 5Β³ = 125
  3. Answer = 27/125

Answer: 27/125

Example: A recipe uses 3/5 cup of sugar, tripled β†’ total = 27/125 cups.

6) Which is larger: 2⁷ or 3⁡?
Estimation & Comparison

Solution:

  1. 2⁷ = 2Γ—2Γ—2Γ—2Γ—2Γ—2Γ—2 = 128
  2. 3⁡ = 3Γ—3Γ—3Γ—3Γ—3 = 243

Answer: 3⁡ = 243 is larger

Example: Two types of investments: doubling weekly vs tripling every week β†’ tripling grows faster.

7) Write as single power: 5Β³ Γ— 5⁴
Same Base, Add Exponents

Solution:

  1. Same base 5 β†’ add exponents: 3+4 = 7
  2. 5Β³Γ—5⁴ = 5⁷

Answer: 57

Example: 5 packs of pencils stacked 3 then 4 β†’ total stack power = 5⁷ pencils.

8) Simplify: 10⁡ ÷ 10²
Subtract Exponents

Solution: 10⁡ Γ· 10Β² = 10^(5βˆ’2) = 10Β³ = 1000

9) Evaluate: (2³)⁴
Power of a Power

Solution:

  1. (2Β³)⁴ = 2^(3Γ—4) = 2ΒΉΒ²
  2. 2ΒΉΒ² = 4096

Answer: 4096

10) Write with positive exponent: 5⁻³
Negative Exponent β†’ Reciprocal

Solution: 5⁻³ = 1/5³ = 1/125

11) Evaluate: (-2)⁡

Step: (-2)⁡ = -32

Odd exponent keeps sign negative.

12) A gamer’s followers triple weekly. Start 2 β†’ 4 weeks?

Solution: 2 Γ— 3⁴ = 2 Γ— 81 = 162 followers

Exponential growth vs linear addition: addition would give 2+3+3+3+3=14 (wrong).

13) File sizes: 1 KB = 2¹⁰ bytes. 8 KB?

8 Γ— 2¹⁰ = 2Β³ Γ— 2¹⁰ = 2ΒΉΒ³ = 8192 bytes

14) Rewrite as a power of 6: 2Β³ Γ— 3Β³

(2Γ—3)Β³ = 6Β³

15) Express 0.001 as a power of 10

0.001 = 1/1000 = 10⁻³

16) Simplify: (5Β² Γ— 5⁻⁴) Γ· 5⁻³

5Β² Γ— 5⁻⁴ = 5⁻²

5⁻² Γ· 5⁻³ = 5⁻² Γ— 5Β³ = 5ΒΉ = 5

17) Express as single power: (7/2)Β³ Γ— (7/2)⁴

(7/2)Β³ Γ— (7/2)⁴ = (7/2)⁷

18) Evaluate: (0.2)Β³ exactly as fraction

0.2 = 1/5 β†’ (1/5)Β³ = 1/125

19) Area of square side 3xΒ²

Area = (3xΒ²)Β² = 3Β² Γ— (xΒ²)Β² = 9x⁴

20) Money: Tsh 50,000 grows 10% per year for 3 years

50,000 Γ— (1.10)Β³ = 50,000 Γ— 1.331 = 66,550

21) Chessboard grains: 1 grain on first, double each square β†’ 8th square?

Square k has 2^(k-1) grains β†’ 2^7 = 128 grains

22) Express 81 as powers of two different bases

81 = 3⁴ = 9²

23) Simplify: (4Β³ Γ— 2⁢) Γ· 8Β²

4Β³ = 2⁢, 8Β² = 2⁢ β†’ (2⁢ Γ— 2⁢)/2⁢ = 2⁢ = 64

24) True/False: (ab)Β³ = aΒ³ + bΒ³

False. Correct: (ab)Β³ = aΒ³bΒ³

25) Evaluate: (-1/2)⁡

(-1/2)⁡ = -1/32

26) Express 0.04 as power of 2 and 5

0.04 = 4/100 = 2²/100 = 2²/(2²·5²) = 5⁻²

Also 2⁰·5⁻²

27) Simplify: (3xΒ²)Β³ Γ— x⁻⁴

3Β³ Γ— (xΒ²)Β³ Γ— x⁻⁴ = 27 x⁢ Γ— x⁻⁴ = 27xΒ²

28) Compare: doubling vs linear +3 after 6 periods

Doubling: 1Γ—2⁢ = 64

Linear: 1 + 6Γ—3 = 19 β†’ Exponential dominates

29) Write as single power: (10⁻¹)⁴

(10⁻¹)⁴ = 10⁻⁴ = 0.0001

30) Rewrite as power with base 27: 3⁹

27 = 3Β³ β†’ 3⁹ = (3Β³)Β³ = 27Β³

Mathematics Questions

Mathematics Questions

  1. Write each of the following numbers without radicals:
    1. √900
    2. √160000
    3. ∛(8 Γ— 27 Γ— 5Β³)
  2. Write each of the following numbers using a radical sign:
    1. 71/2
    2. 192/3
    3. 21/3
  3. Simplify each of the following expressions:
    1. √675 + √75
    2. √1024 + √4
    3. ∛8 + ∛64
    4. √175 + √28 - √63
    5. √(1/2) + 2√(1/2) - √(1/8)
    6. √1000 + √40 - √63
    7. (√25 Γ— √6)
    8. (√75 Γ— √3)
  4. Simplify each of the following radicals by making the number in the radical sign as small as possible:
    1. √50
    2. √375
    3. √125
    4. ∛250
    5. √4096
    6. √1024
    7. √729
    8. ∛625
    9. √1296
    10. √3000
  5. Simplify each of the following radicals:
    1. √(4yΒ³)
    2. √(8y mΒ³)
    3. √(24yΒ³)
    4. √(xyΒ³)
    5. √(729aΒ²bΒ²cΒ²)
  6. Rationalize the denominator in each of the following expressions and simplify:
    1. 1 / √3
    2. 1 / (√2 + 1)
    3. 1 / (√3 + 1)
    4. 1 / (√3 + √2)
    5. (2 / (√5 - 1))
    6. (√3 + 1) / (2√3)
    7. (√6 + 4) / (√6 + √2)
  7. Simplify: ((√2 + √3) / (√3 - √2)) + ((√2 + √3) / (√3 - √2))
  8. Expand each of the following:
    1. (√3 - 1)Β²
    2. (√5 - √2)(√3 - 2)
    3. (√3 + 1)Β²
  9. Find the square root of each of the following:
    1. 2916
    2. 5625
    3. 0.25
  10. Use mathematical tables to evaluate each of the following:
    1. √1256
    2. √0.0015
    3. √256789
    4. √75
    5. √0.009
  11. Given that 1/f = 1/u + 1/v, write v as the subject of the formula.
  12. If vΒ² + uΒ² - 2us = 0, write u as the subject of the formula.
  13. Given the formula s = ut + (3/4)atΒ², express u in terms of other letters.
  14. If s = u + at, express u in terms of a, l, and s.
  15. A formula connecting u, v, and f for a spherical mirror is 1/f = 1/u + 1/v. Calculate the value of v when f = 8.1 and u = 5.4.
Math Solutions - Radicals & Simplifications

Mathematics - Detailed Answers with Notes & Examples

1. Write each of the following numbers without radicals:

  1. √900
  2. √160000
  3. βˆ›(8 Γ— 27 Γ— 5Β³)

(a) √900 = 30 (√ symbol = square root, meaning "what number multiplied by itself gives 900")
Explanation: 30 Γ— 30 = 900 β†’ √900 = 30

(b) √160000 = 400 (Square root of 160000 gives side of square with area 160000)
Explanation: 400 Γ— 400 = 160000

(c) βˆ›(8 Γ— 27 Γ— 5Β³)

  • 8 = 2Β³
  • 27 = 3Β³
  • 5Β³ is already a cube
  • Multiply: 2Β³ Γ— 3Β³ Γ— 5Β³ = (2Γ—3Γ—5)Β³ = 30Β³

βˆ›(30Β³) = 30 (βˆ› symbol = cube root, finding the number that when cubed gives the volume)

Real-life example:
- Square roots: finding the side of a square when area known. Example: football pitch of 900 mΒ² β†’ side = 30 m.
- Cube roots: finding edge of a cube when volume known. Example: sugar cube box 27000 cmΒ³ β†’ side = 30 cm.

2. Write each of the following numbers using a radical sign:

  1. 71/2
  2. 192/3
  3. 21/3

(a) 71/2 = √7 (Fractional exponent 1/2 = square root)

(b) 192/3 = βˆ›(19Β²) = βˆ›361 (2/3 exponent = square then cube root)

(c) 21/3 = βˆ›2 (1/3 exponent = cube root)

Real-life example:
- Powers with fractions = roots. Example: plant growth √time β†’ height proportional to 1/2 power.
- Baker splitting ingredients for cube β†’ cube root used to maintain volume ratios.

3. Simplify each of the following expressions:

  1. √675 + √75
  2. √1024 + √4
  3. βˆ›8 + βˆ›64
  4. √175 + √28 - √63

(a) √675 + √75

  • √675 = √(225Γ—3) = 15√3
  • √75 = √(25Γ—3) = 5√3
  • Total = (15 + 5)√3 = 20√3

(b) √1024 + √4 = 32 + 2 = 34

(c) βˆ›8 + βˆ›64 = 2 + 4 = 6

(d) √175 + √28 - √63

  • √175 = √(25Γ—7) = 5√7
  • √28 = √(4Γ—7) = 2√7
  • √63 = √(9Γ—7) = 3√7
  • Total = (5 + 2 - 3)√7 = 4√7
Real-life example:
- Grouping similar terms like grouping fruits. 675 mangoes β†’ 225Γ—3 = 15√3 per box.
- Farmers measure plots using √ forms to simplify area.

4. Simplify each of the following radicals (make the number inside as small as possible):

  1. √50
  2. √375
  3. √125
  4. βˆ›250
  5. √4096
  6. √1024
  7. √729
  8. βˆ›625
  9. √1296
  10. √3000

(a) √50 = √(25Γ—2) = 5√2

(b) √375 = √(25Γ—15) = 5√15

(c) √125 = √(25Γ—5) = 5√5

(d) βˆ›250 = βˆ›(125Γ—2) = 5βˆ›2

(e) √4096 = 64

(f) √1024 = 32

(g) √729 = 27

(h) βˆ›625 = βˆ›(125Γ—5) = 5βˆ›5

(i) √1296 = 36

(j) √3000 = √(100Γ—30) = 10√30

Real-life example:
- Simplifying units: 50 coins β†’ 5 coins of 10 each = 5√2.
- Builders measure tiles: √50 β†’ 5√2 units for easier cutting and allocation.
Math Solutions - GPT Refusa

Mathematics - Detailed Step-by-Step Solutions (GPT Refusa)

5. Simplify each radical
  1. √(4y³)

    Step 1: Factorize inside radical β†’ 4yΒ³ = 4Β·yΒ²Β·y

    Step 2: Apply √(aΒ·b) = √a Β· √b β†’ √(4Β·yΒ²Β·y) = √4 Β· √(yΒ²) Β· √y = 2y√y

    Example: Imagine 4 boxes each containing yΒ² items; grouping yΒ² together simplifies counting.

  2. √(8y m³)

    Step 1: Factorize β†’ 8y mΒ³ = 4Β·2Β·yΒ·mΒ²Β·m

    Step 2: Simplify β†’ √(4Β·2Β·yΒ·mΒ²Β·m) = √4 Β· √mΒ² Β· √(2ym) = 2m√(2ym)

    Real-life: Counting 8 items distributed into 4 equal parts makes it easier to handle.

  3. √(24y³) = √(4·6·y²·y) = 2y√(6y)
  4. √(xy³) = √(y²·xy) = y√(xy)
  5. √(729a²b²c²) = √(27²·a²b²c²) = 27abc
6. Rationalize denominators
  1. 1/√3 β†’ multiply numerator & denominator by √3 β†’ √3/3

    Explanation: We remove root from denominator so division is easier.

  2. 1/(√2+1) β†’ multiply top & bottom by (√2-1) β†’ (√2-1)/(2-1) = √2-1
  3. 1/(√3+1) β†’ multiply by (√3-1) β†’ (√3-1)/(3-1) = (√3-1)/2
  4. 1/(√3+√2) β†’ multiply by (√3-√2) β†’ (√3-√2)/(3-2) = √3-√2
  5. 2/(√5-1) β†’ multiply by (√5+1) β†’ (2(√5+1))/(5-1) = (2√5+2)/4 = (√5+1)/2
  6. (√3+1)/(2√3) β†’ split β†’ (√3)/(2√3) + 1/(2√3) = 1/2 + √3/6

Tip: Think of rationalizing as converting "complex units" into simple, countable units, like changing √kg into kg for weighing.

7. Simplify ((√2+√3)/(√3-√2)) + ((√2+√3)/(√3-√2))

Step 1: Combine like terms β†’ 2 Γ— (√2+√3)/(√3-√2)

Step 2: Multiply numerator and denominator by conjugate (√3+√2)

Step 3: Expand numerator β†’ (√2+√3)(√3+√2) = 2 + 3 + 2√6 = 5 + 2√6

Step 4: Divide by denominator β†’ (3-2) = 1

Step 5: Multiply by 2 β†’ 2 Γ— (5+2√6) = 10 + 4√6

Example: Like paying twice for an item β†’ multiply total cost by 2 rather than adding separately.

8. Expand radicals
  1. (√3 - 1)Β² β†’ (√3)Β² - 2·√3Β·1 + 1Β² = 3 - 2√3 + 1 = 4 - 2√3
  2. (√5 - √2)(√3 - 2) β†’ multiply each β†’ √15 - 2√5 - √6 + 2√2
  3. (√3 + 1)² = 3 + 2√3 + 1 = 4 + 2√3

Tip: Expanding radicals is like breaking down total expenses into individual items to understand full costs.

9. Find square roots

√2916 β†’ Factorize: 2916 = 2Β² Β· 3⁢ β†’ √(2Β²Β·3⁢) = 2Β·3Β³ = 2Β·27 = 54

√5625 = 75

√0.25 = 0.5

Real-life: Side of square plot with area 2916 mΒ² β†’ side = √2916 = 54 m

10. Approximate using tables

√1256 β‰ˆ 35.46 β†’ use square root table or interpolation

√0.0015 β‰ˆ 0.0387

√256789 β‰ˆ 506.74

√75 β‰ˆ 8.66

√0.009 β‰ˆ 0.0949

Tip: Approximation is used when exact calculation is hard, similar to estimating weight using scale readings.

11. Mirror formula: 1/f = 1/u + 1/v β†’ v as subject

Step 1: Start from 1/f = 1/u + 1/v

Step 2: Subtract 1/u β†’ 1/v = 1/f - 1/u

Step 3: Invert both sides β†’ v = 1 / (1/f - 1/u)

Example: Like calculating distance of image from mirror when object and focal length are known.

12. Solve vΒ² + uΒ² - 2us = 0 β†’ u as subject

vΒ² + uΒ² - 2us = 0 β†’ uΒ² - 2us + vΒ² = 0 β†’ Quadratic in u

Use formula β†’ u = [2s Β± √((2s)Β² - 4Β·1Β·vΒ²)] / 2 = s Β± √(sΒ² - vΒ²)

Example: Like finding initial velocity given displacement and final velocity in physics problem.

13. s = ut + (3/4)atΒ² β†’ u as subject

Step 1: s = ut + (3/4)atΒ² β†’ isolate ut β†’ ut = s - (3/4)atΒ²

Step 2: Divide both sides by t β†’ u = [s - (3/4)atΒ²] / t

Example: Like finding starting speed of a car if distance and acceleration are known.

14. s = u + at β†’ u as subject

Step 1: s = u + at β†’ isolate u β†’ u = s - at

Example: Determine initial speed if final speed after time t is known.

15. Mirror problem: f = 8.1, u = 5.4 β†’ find v

Step 1: 1/v = 1/f - 1/u = 1/8.1 - 1/5.4

Step 2: Compute decimals β†’ 0.1235 - 0.1852 = -0.0617

Step 3: Invert β†’ v = 1 / -0.0617 β‰ˆ -16.2

Note: Negative value indicates virtual image (like plane mirror, behind mirror).

Complete Math Notes - Radicals & Powers

Complete Math Notes: Radicals, Roots & Powers

1. Introduction to Radicals

Radicals represent roots of numbers. Most common:

  • Square root: √a = b β†’ bΒ² = a
  • Cube root: βˆ›a = b β†’ bΒ³ = a
  • n-th root: ⁿ√a = b β†’ bⁿ = a
  • Exponential form: a^(m/n) = ⁿ√(a^m)
Shortcut for simplifying radicals:
√(aΓ—b) = √a Γ— √b, βˆ›(aΓ—b) = βˆ›a Γ— βˆ›b
√(xΒ²y) = x√y, βˆ›(xΒ³y) = xβˆ›y

Example: √50 = √(25Γ—2) = 5√2, βˆ›250 = βˆ›(125Γ—2) = 5βˆ›2

2. Powers & Fractional Exponents
  • a^(1/2) = √a
  • a^(2/3) = βˆ›(aΒ²)
  • a^(m/n) = ⁿ√(a^m)
  • Multiplication: a^m Γ— a^n = a^(m+n)
  • Division: a^m Γ· a^n = a^(m-n)
  • Power of a power: (a^m)^n = a^(mΓ—n)
Example: 8^(2/3) = βˆ›(8Β²) = βˆ›64 = 4
3. Simplifying Radicals
  1. Factor number inside radical into square/cube factors
  2. Take perfect squares/cubes outside the radical
  3. Multiply coefficients outside radical

Examples: √72 = √(36Γ—2) = 6√2, βˆ›54 = βˆ›(27Γ—2) = 3βˆ›2

Tip: Always reduce the number inside the radical as much as possible.
4. Rationalizing Denominators
  • 1/√a = √a / a
  • 1/(√a + √b) = (√a - √b)/(a-b)
  • 1/(√a - √b) = (√a + √b)/(a-b)
  • Use conjugates to remove radicals from denominator

Example: 1/(√3+1) = (√3-1)/2

5. Expansions & Simplifications
  • (a+b)Β² = aΒ² + 2ab + bΒ²
  • (a-b)Β² = aΒ² - 2ab + bΒ²
  • (a+b)(c+d) = ac + ad + bc + bd
  • (√a + √b)Β² = a + 2√(ab) + b
  • (√a - √b)Β² = a - 2√(ab) + b

Example: (√3 + 1)² = 3 + 2√3 + 1 = 4 + 2√3

6. Important Formulas
  • Mirror formula: 1/f = 1/u + 1/v β†’ v = 1/(1/f - 1/u)
  • Quadratic: axΒ² + bx + c = 0 β†’ x = [-b Β± √(bΒ²-4ac)]/(2a)
  • Motion formulas:
    • s = ut + 1/2 atΒ² β†’ u = (s - 1/2 atΒ²)/t
    • s = u + at β†’ u = s - at
7. Step-by-Step Methods & Tips
  • Factor numbers to extract perfect squares/cubes
  • Combine like radicals: √2 + 3√2 = 4√2
  • Rationalize denominators using conjugates
  • Expand using (a+b)Β², (a-b)Β², (a+b)(c+d)
  • Use tables or approximations for non-perfect squares
  • Always verify by squaring/cubing your answer
8. Practice Questions
  1. Simplify √225, βˆ›343, 16^(1/2), 27^(2/3)
  2. Simplify √72 + √18
  3. Simplify βˆ›54 + βˆ›16
  4. Rationalize: 1/√7, 1/(√5 + 2)
  5. Expand: (√2 + 3)², (√5 - √3)(√5 + √3)
  6. Given s = ut + 1/2 atΒ², find u in terms of s, a, t
  7. Mirror problem: f = 10, u = 6 β†’ find v
  8. Combine: √50 + √18 - √32
  9. βˆ›(8Γ—27Γ—64) β†’ simplify
  10. Simplify √(48x³y²)
9. Real-Life Applications
  • Square roots: calculate sides of squares, areas of rooms
  • Cube roots: volumes of boxes, sugar cubes, tanks
  • Rationalizing: simplifying fractions in recipes, finance
  • Expansions: cost analysis, probability calculations
  • Fractional powers: growth rates, physics formulas, scaling
Complex Math Questions & Solutions

Complex Mathematics - Step by Step Solutions

1. Simplify: √(50) Γ— √(18) Γ· √(8)

Step 1: Combine radicals using multiplication/division rules:

√(50) Γ— √(18) Γ· √(8) = √(50Γ—18 Γ· 8)

Step 2: Multiply and divide inside the radical:

50Γ—18 = 900, 900 Γ· 8 = 112.5

Step 3: Express 112.5 in simplified radical form:

112.5 = 16Γ—7.03125 (not perfect square), so we approximate:

√112.5 β‰ˆ 10.606 (rounded)

Answer: β‰ˆ 10.61

Note: Using √a Γ— √b = √(aΓ—b), and √a Γ· √b = √(a/b)
Example: - Imagine cutting 50 mΒ² and 18 mΒ² tiles and trying to combine them into square patches divided by 8 units. The resulting patch side β‰ˆ 10.61 m.

2. Solve for x: 2√(x+5) - 3 = 7

Step 1: Isolate the radical:

2√(x+5) = 7 + 3 = 10

Step 2: Divide both sides by 2:

√(x+5) = 5

Step 3: Square both sides to remove the radical:

x+5 = 25

Step 4: Solve for x:

x = 25 - 5 = 20

Note: Always check if squaring introduces extraneous solutions. Here it does not.
Example: - A gardener wants √(x+5) meters of rope to reach a height minus 3 meters adjustment = 7 meters. Step by step, the rope length x = 20 meters.

3. Simplify: (3x√2 + 4√8) - (2√18 - x√2)

Step 1: Simplify individual radicals:

  • 4√8 = 4√(4Γ—2) = 4Γ—2√2 = 8√2
  • 2√18 = 2√(9Γ—2) = 2Γ—3√2 = 6√2

Step 2: Rewrite the expression:

(3x√2 + 8√2) - (6√2 - x√2) = 3x√2 + 8√2 - 6√2 + x√2

Step 3: Combine like terms:

(3x + x)√2 + (8 - 6)√2 = 4x√2 + 2√2

Answer: 4x√2 + 2√2 = (4x + 2)√2

Combine coefficients of √2 carefully; group variable and constants separately.
Example: - A construction worker combining lengths of rods in √2 units: 3x√2 + 4√8 rods, subtracting other sections, final length = (4x+2)√2 units.

4. Evaluate: βˆ›(54 Γ— 16) Γ· βˆ›2 + βˆ›(32)

Step 1: Combine first radicals using division:

βˆ›(54Γ—16) Γ· βˆ›2 = βˆ›((54Γ—16)/2) = βˆ›(864/2) = βˆ›432

Step 2: Simplify βˆ›432:

432 = 8 Γ— 54 = 2Β³ Γ— 2 Γ— 27 = 2Β³ Γ— 2 Γ— 3Β³ = 2⁴ Γ— 3Β³

Take cube root:

βˆ›(2⁴ Γ— 3Β³) = βˆ›(2Β³ Γ— 3Β³ Γ— 2) = βˆ›(2Β³Γ—3Β³) Γ— βˆ›2 = 6βˆ›2

Step 3: Add βˆ›32:

βˆ›32 = βˆ›(2⁡) = βˆ›(2Β³ Γ— 2Β²) = 2βˆ›4

Step 4: Add results:

6βˆ›2 + 2βˆ›4 (cannot combine further as radicals inside different)

Answer: 6βˆ›2 + 2βˆ›4

Cube roots require factorization into perfect cubes Γ— remaining radical to simplify.
Example: - Packing cubic boxes with volume 54Γ—16 and adding 32 units boxes, the final edge in cube roots = 6βˆ›2 + 2βˆ›4 units.

Reference Book: N/A

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