Lesson 6: Indices

Square and cube: When the index is 2, the power is called a square (7² = 49), and when the index is 3, it is called a cube (4³ = 64).

Zero and negative indices: The power 0 of any non-zero number is 1 (9⁰ = 1). A negative index means the reciprocal of the number with the same index made positive: 10⁻² = 110² = 1100.

In the figure: The yellow chips show the base 2 five times, and the blue number 5 below them is the index. So 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.

Rule 1: Identifying the base and the index — The big number below is the base and the small number above is the index. Take the base and multiply it as many times as the index says.

• Easy: 6⁵ = 6 × 6 × 6 × 6 × 6 = 7776. Base 6, index 5.
• Medium: 12³ = 12 × 12 × 12 = 1728. Base 12, index 3.
• Exam type: Write 108 as a product of prime factors: 108 = 2 × 2 × 3 × 3 × 3 = 2² × 3³. Each different base has its own index.

Rule 2: Square and cube — When the index is 2, multiply the number by itself once; when the index is 3, write the number three times and multiply.

• Easy: 13² = 13 × 13 = 169.
• Medium: 6³ = 6 × 6 × 6 = 216.
• Exam type: A square with sides of 20 cm has area 20² = 400 sq cm, and a cube with edges of 20 cm has volume 20³ = 8000 cu cm.

Rule 3: Power of a negative number — When the index is even, the answer is positive; when the index is odd, the answer is negative.

• Easy: (−3)² = (−3) × (−3) = 9.
• Medium: (−2)⁵ = (−2) × (−2) × (−2) × (−2) × (−2) = −32. The index 5 is odd, so the answer is negative.
• Exam type: (−1)²⁵ + (−1)⁴⁰ = (−1) + 1 = 0. An odd power of (−1) is −1 and an even power is 1.

In the figure: Start from 10³ = 1000. At each step down, divide by 10 and the index goes down by 1: 10² = 100, 10¹ = 10, 10⁰ = 1, then 10⁻¹ = 110 and 10⁻² = 1100.

Rule 4: Zero index — The power 0 of any non-zero number is 1, because a number divided by itself gives 1.

• Easy: 17⁰ = 1.
• Medium: 2⁶ ÷ 2⁶ = 64 ÷ 64 = 1. The index is 6 − 6 = 0, so 2⁰ = 1.
• Exam type: 7⁰ + 9⁰ − 4⁰ = 1 + 1 − 1 = 1.

Rule 5: Negative index — When the index is negative, take the reciprocal of the base (1 ÷ base) and make the index positive.

• Easy: 10⁻³ = 110³ = 11000.
• Medium: 4⁻² = 14² = 116.
• Exam type: 12⁻⁵ = 2⁵ = 32. A negative index in the denominator moves up to the numerator and becomes positive.

Rule 6: Negative index of a fraction — When the index of a fraction is negative, turn the fraction upside down (swap numerator and denominator) and make the index positive.

• Easy: (23)⁻¹ = 32.
• Medium: (34)⁻² = (43)² = 169.
• Exam type: 5⁻² = 125 and (15)² = 125. Both come out the same, so taking the reciprocal 15 of 5 and making the index positive is correct.

Spot the difference

In the figure: Put 3 chips of 4 together with 2 chips of 4 and you get 3 + 2 = 5 chips. So 4³ × 4² = 4⁵.

Rule 1: Multiplication (same base) — Keep the base and add the indices.

• Easy: 4³ × 4² = 4^{3 + 2} = 4⁵ = 1024.
• Medium: (−2)³ × (−2)⁴ = (−2)^{3 + 4} = (−2)⁷ = −128.
• Exam type: In 4² × 2³ the bases are different. 4 = 2², so 4² = 2⁴ and 2⁴ × 2³ = 2⁷ = 128.

In the figure: On top there are 5 chips of 5 and below there are 2 chips of 5. 2 pairs cancel and 3 chips are left. So 5⁵ ÷ 5² = 5^{5 − 2} = 5³.

Rule 2: Division (same base) — Keep the base and subtract the index of the divisor from the index of the dividend.

• Easy: 5⁵ ÷ 5² = 5^{5 − 2} = 5³ = 125.
• Medium: 7³ ÷ 7⁵ = 7^{3 − 5} = 7⁻² = 149. When the index of the divisor is bigger, the answer comes with a negative index.
• Exam type: (−3)⁷ ÷ (−3)⁴ = (−3)^{7 − 4} = (−3)³ = −27.

Rule 3: Multiplication and division with negative indices — The rule stays the same; just take care of the signs while adding or subtracting the indices.

• Easy: 2⁻³ × 2⁵ = 2^{−3 + 5} = 2² = 4.
• Medium: 2⁻² × 2⁻³ = 2⁻⁵ = 132.
• Exam type: 3² ÷ 3⁻³ = 3^{2 − (−3)} = 3⁵ = 243.

Spot the difference

In the figure: There are 3 such groups of 2³, and each group has 2 three times. In all, 2 appears 3 × 3 = 9 times. So (2³)³ = 2⁹.

Rule 1: Power of a power — Keep the base and multiply the indices with each other.

• Easy: (3²)⁴ = 3^{2 × 4} = 3⁸ = 6561.
• Medium: (5⁻²)³ = 5^{−2 × 3} = 5⁻⁶ = 115625.
• Exam type: (3²)³ × 3⁻⁴ = 3⁶ × 3⁻⁴ = 3² = 9.

In the figure: (2 × 5) is written three times. Gather all the 2s together and all the 5s together and you get 2³ × 5³ = 8 × 125 = 1000. So (2 × 5)³ = 2³ × 5³.

Rule 2: Power of a product — Give that power to every factor of the product and then multiply.

• Easy: (2 × 5)³ = 2³ × 5³ = 8 × 125 = 1000.
• Medium: (5 × 4)² = 5² × 4² = 25 × 16 = 400.
• Exam type: When the indices are the same, the bases can be multiplied first: 2⁴ × 5⁴ = (2 × 5)⁴ = 10⁴ = 10000.

Rule 3: Power of a quotient — Give that power to both the numerator and the denominator.

• Easy: (34)² = 3 × 34 × 4 = 916.
• Medium: (25)³ = 2 × 2 × 25 × 5 × 5 = 8125.
• Exam type: (96)² in two ways: first 96 = 32, so (32)² = 94; or directly 8136 = 94 (divide both by 9).

Rule 4: Writing a negative index as a positive index — When a negative index comes up, take the reciprocal of the base and make the index positive. This works for powers, products and fractions alike.

• Easy: (37)⁻² = (73)² = 499.
• Medium: (3 × 5)⁻² = 15⁻² = 115² = 1225.
• Exam type: Write a negative index in the form of a positive index: 2⁻⁶ = 12⁶ = 164.

Spot the difference

In the figure: In 4700000, move the decimal point 6 places to the left and you get 4.7, so 4700000 = 4.7 × 10⁶. In 0.00032, move the decimal point 4 places to the right and you get 3.2, so 0.00032 = 3.2 × 10⁻⁴. The red dot is the real point and the violet ring shows the new place.

Rule 1: A big number in standard form — The number of places the point moves to the left is the positive index of 10.

• Easy: 4700000 = 4.7 × 10⁶ (the point moves 6 places to the left).
• Medium: 150000000 = 1.5 × 10⁸ (the point moves 8 places to the left).
• Exam type: 36000000 = 3.6 × 10⁷ (the point moves 7 places to the left).

Rule 2: An ordinary number from standard form — Move the point to the right by as many places as the index of 10; fill the empty places with zeros.

• Easy: 5.2 × 10⁴ = 52000.
• Medium: 8.03 × 10⁵ = 803000.
• Exam type: 2.5 × 10⁷ = 25000000.

Rule 3: A small number in standard form — The number of places the point moves to the right is the negative index of 10.

• Easy: 0.0004 = 4 × 10⁻⁴ (the point moves 4 places to the right).
• Medium: 0.00032 = 3.2 × 10⁻⁴ (the point moves 4 places to the right).
• Exam type: 0.0000025 = 2.5 × 10⁻⁶ (the point moves 6 places to the right).

Rule 4: Comparing numbers in standard form — First look at the index of 10; the number with the bigger index is bigger. If the indices are the same, compare the first parts.

• Easy: 6 × 10⁵ and 2 × 10⁷: the index 7 is bigger, so 2 × 10⁷ is bigger.
• Medium: 4.8 × 10⁶ and 3.9 × 10⁶: the indices are the same and 4.8 > 3.9, so 4.8 × 10⁶ is bigger.
• Exam type: 5 × 10⁻³ and 2 × 10⁻⁵: −3 is bigger than −5, so 5 × 10⁻³ (that is 0.005) is bigger; the other number is 0.00002.

Spot the difference

In the figure: A square is made of 11 × 11 = 121 tiles. Its side is 11, so √121 = 11.

Rule 1: Square and square root — The number whose square is the given number is its square root. Every positive perfect square has two roots, one positive and one negative; the sign √ shows only the positive root.

• Easy: 13² = 169, so √169 = 13.
• Medium: 13 × 13 = 169 and (−13) × (−13) = 169. So the roots of 169 are +13 and −13.
• Exam type: 16² = 256, so √256 = 16 and the roots of 256 are ±16.

In the figure: Dividing 196 again and again by primes gives 2, 2, 7, 7, and the quotients come out as 98, 49, 7, 1. The equal factors form two pairs, and taking one factor from each pair gives 2 × 7 = 14. So √196 = 14.

Rule 2: Prime factor method — Write the number as a product of prime factors, make pairs of equal factors, pick one factor out of every pair and multiply those.

• Easy: 196 = 2 × 2 × 7 × 7, so √196 = 2 × 7 = 14.
• Medium: 576 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 = 2⁶ × 3². There are 3 pairs of 2 and 1 pair of 3. So √576 = 2 × 2 × 2 × 3 = 24.
• Exam type: 252 = 2 × 2 × 3 × 3 × 7. The 7 has no partner, so 252 is not a perfect square. To complete the pair, multiply by 7: 252 × 7 = 1764 = 42², so √1764 = 42.

In the figure: On the top line are 36, 40 and 49, and on the bottom line are their roots 6, √40 and 7 (6² = 36 and 7² = 49). 40 lies between 36 and 49, so √40 lies between 6 and 7.

Rule 3: Square root of a number that is not a perfect square — Find the nearest perfect squares on both sides of the given number. The square root lies between their square roots.

• Easy: 36 < 40 < 49, so 6 < √40 < 7.
• Medium: 6.3² = 39.69 and 6.4² = 40.96. 40 lies between 39.69 and 40.96, so √40 lies between 6.3 and 6.4.
• Exam type: 81 < 90 < 100, so 9 < √90 < 10.

In the figure: Make pairs of digits of 5476 from the right: 54 and 76. The nearest square below 54 is 49 (7 × 7 = 49), so the first digit is 7. The remainder is 5 and bringing down 76 makes 576. The divisor is 7 × 2 = 14; 144 × 4 = 576 and the remainder is 0. So √5476 = 74.

Rule 4: Division method — Make pairs of digits, take the square nearest to the first pair, bring down the next pair to the remainder, and double the divisor to find the next digit. For a decimal number, make pairs on both sides of the point.

• Easy: √5476 = 74 (as in the figure above). Check: 74 × 74 = 5476.
• Medium: √70.56: the pairs are 70 and 56 (after the point). 8 × 8 = 64, remainder 6, bring down 56 to get 656. The divisor is 8 × 2 = 16; 164 × 4 = 656 and the remainder is 0. So √70.56 = 8.4.
• Exam type: √1521: the pairs are 15 and 21. 3 × 3 = 9, remainder 6, bring down 21 to get 621. The divisor is 3 × 2 = 6; 69 × 9 = 621 and the remainder is 0. So √1521 = 39.

Spot the difference