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State Board · Class 8 · Mathematics

Lesson 3: Indices and Cube root

5 concepts · 40 practice questions with answers · Free to read

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Laws of indices

From daily life

A bacterium splits into 2 every hour. After 5 hours there are 2 × 2 × 2 × 2 × 2 = 32 bacteria. Writing this as 25 is much shorter, and the laws of indices let us multiply and divide such numbers without writing them out.

Simple definition

In an, a is the base (पाया) and n is the index (घातांक; power). an means a multiplied by itself n times.

Rule 1: Laws of indices — am × an = am+n. am ÷ an = am−n. (am)n = amn. (ab)m = ambm. a0 = 1. a−m = 1am. (a/b)−m = (b/a)m.

25baseindex22222

In the figure : In the figure 25 = 2 × 2 × 2 × 2 × 2: the base 2 is written 5 times, and 5 is the index.

4443×44245

In the figure : In the figure product law: 3 fours and 2 fours make 5 fours, so 43 × 42 = 43+2 = 45.

66666666666−2

In the figure : In the figure quotient law: 4 sixes above cancel 4 of the 6 sixes below, leaving 2 sixes below: 64 ÷ 66 = 64−6 = 6−2 = 136.

2222242222242222243 rows212=

In the figure : In the figure power of a power: 3 rows of 24 hold 12 twos, so (24)3 = 24×3 = 212.

33=27÷332=9÷331=3÷330=1÷33−1=13÷33−2=19

In the figure : In the figure each row down divides by the base: 31 = 3, 30 = 1, 3−1 = 13 and 3−2 = 19.

  • Easy : Easy: 53 × 54 = 57. And 100 = 1.
  • Medium : Medium: 74 ÷ 77 = 7−3 = 1343. And (23)4 = 212.
  • Exam-type : Exam: (2/3)−2 = (3/2)2 = 94. And (2 × 5)3 = 23 × 53 = 8 × 125 = 1000.
Remember

Same base: multiply → add indices; divide → subtract; power of a power → multiply.

Common mistake

Writing 23 × 24 = 47. The base stays 2: the answer is 27.

Shortcut

Memory clue: "गुणाकार = बेरीज, भागाकार = वजाबाकी".

Spot the difference

LawResult
am × anam+n
am ÷ anam−n
(am)namn
a01

Practice questions

Easy
  1. 1What are the base and index in 34?
  2. 2Simplify 62 × 65.
  3. 3Simplify 98 ÷ 95.
Medium
  1. 4Simplify (32)3.
  2. 5Value of 120?
  3. 6Value of 4−2?
Challenging
  1. 7Simplify (25 × 2−2) ÷ 21.
  2. 8Why is a0 = 1?
Show answers (try them yourself first)
  1. 1Base 3, index 4.
  2. 267.
  3. 393 = 729.
  4. 436 = 729.
  5. 51.
  6. 6116.
  7. 723−1 = 22 = 4.
  8. 8am ÷ am = am−m = a0 and any number divided by itself is 1.

Roots as indices (index 1/n)

From daily life

If a square garden has area 49 m2, its side is the number that squares to 49. The idea of the index 12 means exactly this: undo the square.

Simple definition

The nth root (nवे मूळ) of a is written a1/n. It is the number that, multiplied n times, gives a.

Rule 1: Roots as indices — 491/2 = 7 because 72 = 49. 1251/3 = 5 because 53 = 125. 811/4 = 3 because 34 = 81. 321/5 = 2 because 25 = 32. If 101/5 = x then x5 = 10.

index133437inputoutput73=343

In the figure : In the figure the cube root as an index: 3431/3 = 7 because 73 = 343. The index 13 undoes the cube.

41×4162×4643×42564number of factors

In the figure : In the figure the nth root: 256 is made of four 4s multiplied together, so 2561/4 = 4. In general a1/n is the number that is multiplied n times to give a.

  • Easy : Easy: 641/2 = 8 and 271/3 = 3.
  • Medium : Medium: 6251/4 = 5 because 5 × 5 × 5 × 5 = 625.
  • Exam-type : Exam: Write in index form: the 6th root of 11 is 111/6; the square root of 200 is 2001/2.
Remember

a1/n is the nth root. The index 1/2 is the square root; 1/3 is the cube root.

Common mistake

Thinking a1/2 means a ÷ 2. It means the square root of a.

Shortcut

Memory clue: "1/n घातांक = nवे मूळ".

Spot the difference

ExpressionMeaningValue
491/2Square root7
1251/3Cube root5
811/44th root3

Practice questions

Easy
  1. 1Write the square root of 15 as an index.
  2. 2Value of 361/2?
  3. 3Value of 641/3?
Medium
  1. 4Value of 161/4?
  2. 5Value of 2431/5?
  3. 6Write the 9th root of 512 as an index and find it.
Challenging
  1. 7If x = 171/3, what is x3?
  2. 8Why is 10001/3 = 10?
Show answers (try them yourself first)
  1. 1151/2.
  2. 26.
  3. 34.
  4. 42.
  5. 53 because 35 = 243.
  6. 65121/9 = 2.
  7. 717.
  8. 810 × 10 × 10 = 1000.

Rational indices m/n

From daily life

Think of a recipe: "take the cube root first, then square" or "square first, then take the cube root" - both give the same dish. That is a rational index m/n.

Simple definition

For a rational index, am/n = (a1/n)m = (am)1/n: the mth power of the nth root, or the nth root of the mth power.

Rule 1: Evaluating m/n — Take the root first, it keeps numbers small. 82/3 = (81/3)2 = 22 = 4. 163/4 = (161/4)3 = 23 = 8. 253/2 = 53 = 125. 322/5 = 22 = 4. Negative: 8−2/3 = 14.

27power 2729root 39root 33power 29the two routes give the same value

In the figure : In the figure 272/3 two ways: (272)1/3 = 7291/3 = 9 and (271/3)2 = 32 = 9. Both routes give the same value.

  • Easy : Easy: 43/2 = 23 = 8.
  • Medium : Medium: 813/4 = 33 = 27.
  • Exam-type : Exam: 45/2 = 25 = 32, and 9−3/2 = 127.
Remember

am/n: denominator n is the root, numerator m is the power.

Common mistake

Swapping m and n. Check: 82/3 uses the cube root (3), then squares (2).

Shortcut

Memory clue: "छेद = मूळ, अंश = घात".

Spot the difference

ExpressionStepsValue
82/3(∛8)24
163/4(⁴√16)38
253/2(√25)3125

Practice questions

Easy
  1. 1Meaning of am/n?
  2. 2Evaluate 93/2.
  3. 3Evaluate 1252/3.
Medium
  1. 4Evaluate 163/2.
  2. 5Evaluate 642/3.
  3. 6Evaluate 274/3.
Challenging
  1. 7Evaluate 8−2/3.
  2. 8Why do we take the root first?
Show answers (try them yourself first)
  1. 1The mth power of the nth root of a.
  2. 233 = 27.
  3. 352 = 25.
  4. 443 = 64.
  5. 542 = 16.
  6. 634 = 81.
  7. 714.
  8. 8The numbers stay small and easy.

Square roots, cubes and cubes of numbers

From daily life

A cube of sugar with 3 small cubes along each edge holds 3 × 3 × 3 = 27 small cubes. That is why the third power is called a cube. And a square root has two answers: 5 × 5 and (−5) × (−5) both make 25.

Simple definition

The cube (घन) of a is a × a × a = a3. A positive number has two square roots, one positive and one negative.

Rule 1: Square roots and cubes — 49 has the roots 7 and −7 (√49 = 7, −√49 = −7). √0 = 0. Cubes: 73 = 343; 133 = 2197; (−5)3 = −125; (−34)3 = −2764; (2.5)3 = 15.625; (0.3)3 = 0.027.

(−5)5squaresquare25

In the figure : In the figure a positive number has two square roots: 5 and −5 both square to 25, so the roots of 25 are 5 and −5 (√25 = 5, −√25 = −5).

3edge27small cubes

In the figure : In the figure a cube with 3 small cubes along every edge has 3 × 3 × 3 = 33 = 27 small cubes: this is why the third power is called the cube.

(−4)(−4)(−4)××+16−64

In the figure : In the figure cube of a negative number: (−4) × (−4) = 16 and 16 × (−4) = −64, so (−4)3 = −64 (negative).

151 placecube33753 places

In the figure : In the figure decimal places: 1.5 has 1 decimal place, so (1.5)3 = 3.375 has 3 × 1 = 3 decimal places.

  • Easy : Easy: 63 = 216 and (−2)3 = −8.
  • Medium : Medium: √144 = 12, and the other root is −12.
  • Exam-type : Exam: (0.2)3 = 0.008 has 3 decimal places because 0.2 has 1.
Remember

Cube of positive is positive; cube of negative is negative. Decimal places triple.

Common mistake

Saying (−3)3 = 27. It is −27, because an odd number of negatives gives negative.

Shortcut

Memory clue: "ऋणाचा घन ऋणच".

Spot the difference

NumberCube
7343
−5−125
0.30.027

Practice questions

Easy
  1. 1What is the cube of 9?
  2. 2What is the cube of −3?
  3. 3What are the square roots of 64?
Medium
  1. 4Cube of 25?
  2. 5Cube of 1.2?
  3. 6Cube of 0.02?
Challenging
  1. 7Is the cube of a negative number positive?
  2. 8(1.5)3 has how many decimal places?
Show answers (try them yourself first)
  1. 1729.
  2. 2−27.
  3. 38 and −8.
  4. 48125.
  5. 51.728.
  6. 60.000008.
  7. 7No, negative.
  8. 83 × 1 = 3 (3.375).

Cube roots

From daily life

A carpenter has a cubical block of volume 1000 cm3 and wants its edge length. The edge is the number that, multiplied three times, gives 1000. That number is the cube root.

Simple definition

The cube root (घनमूळ) of a is the number whose cube is a. Find it by prime factorisation, grouping equal factors in threes.

Rule 1: Cube root by prime factors — 1000 = 2·2·2·5·5·5, so the cube root is 2 × 5 = 10. 3375 = 3·3·3·5·5·5, so the cube root is 3 × 5 = 15. 1728 = (2·2·2)·(2·2·2)·(3·3·3), root 2 × 2 × 3 = 12. The cube root of a negative number is negative: ∛(−2197) = −13.

1000 = 2 × 2 × 2 × 5 × 5 × 522225555cube root = 2 × 5 = 10

In the figure : In the figure cube root by prime factors: each group of three equal factors gives one factor of the root, 2 × 5 = 10, so the cube root of 1000 is 10.

  • Easy : Easy: ∛343 = 7 because 7 × 7 × 7 = 343.
  • Medium : Medium: ∛2764 = 34 and ∛0.027 = 0.3.
  • Exam-type : Exam: Since ∛729 = 9, ∛0.000729 = 0.09 (6 decimal places become 2).
Remember

Groups of three equal factors. Negative number → negative root.

Common mistake

Taking pairs of factors, as for a square root. For a cube root take triples.

Shortcut

Memory clue: "घनमूळ = तिघांचा गट".

Spot the difference

NumberPrime factorsCube root
10002³ × 5³10
33753³ × 5³15
−2197−(13³)−13

Practice questions

Easy
  1. 1What is a cube root?
  2. 2How do we group prime factors for a cube root?
  3. 3Cube root of 8000?
Medium
  1. 4Cube root of 512?
  2. 5Cube root of −2744?
  3. 6Cube root of 1654?
Challenging
  1. 7Cube root of 0.125?
  2. 8Find the edge of a cube of volume 3375 cm3.
Show answers (try them yourself first)
  1. 1The number whose cube is the given number.
  2. 2In threes.
  3. 320.
  4. 48.
  5. 5−14.
  6. 61654 = 827, root 23.
  7. 70.5.
  8. 815 cm.

Quick revision

  • am × an = am+n; a0 = 1; a-m = 1/am; (am)n = amn.
  • nth root of a = a1/n; am/n = nth root of the mth power = mth power of the nth root.
  • The cube of a negative number is negative; find a cube root by grouping prime factors in threes.
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