Number, Year 10: counting, powers and roots

24

HintEvery sandwich can go with every drink, and each of those pairs can go with every dessert.

The whyThere are 4 × 3 = 12 sandwich-and-drink pairs, and each pair can be finished with either of 2 desserts, so 12 × 2. Multiplying the number of choices at each stage counts every combination without listing them.

The product rule for counting

HintIts name uses the word for the answer to a multiplication.

The whyWith 3 starters and 5 mains there are 3 × 5 = 15 two-course meals, because each starter can be paired with every main. It replaces a long systematic list with one multiplication.

5; 4; 20

HintOnce the first letter is used it cannot be picked again.

The whyAny of the five letters can come first. Whichever is chosen, only four remain for the second place, so the product rule gives 5 × 4. If letters could repeat, it would be 5 × 5 = 25.

10

HintCount the opponents each team has, multiply, then ask how many times each match has been counted.

The whyEach of the 5 teams has 4 opponents, and 5 × 4 = 20. That counts every match twice (A v B and B v A are one match), so halve it: 20 ÷ 2. When order does not matter, the product rule over-counts and you divide.

Each starter can be paired with every one of the main courses

HintPicture a grid: one row per first dish, one column per second dish.

The whyStarter 1 gives 4 different meals, and so do starters 2 and 3, making 4 + 4 + 4 = 3 × 4 = 12. A sample-space grid shows the same thing: rows times columns.

7

HintFind the square numbers on either side of 50.

The why7² = 49 and 8² = 64, and 50 lies between 49 and 64, so √50 lies between 7 and 8. Because 50 is only just above 49, the root is only just above 7 (about 7.07).

About 9.5

HintSee how far 90 is along the gap from 81 to 100.

The why90 is 9 above 81, and the gap from 81 to 100 is 19, so 90 is roughly halfway. That puts the root roughly halfway from 9 to 10. A check: 9.5² = 90.25, very close to 90.

Between 4 and 5

HintList the first few cube numbers until you pass 100.

The why4³ = 64 and 5³ = 125, and 100 lies between them. It is a little past halfway from 64 to 125, so ∛100 is about 4.6.

64, which is a little too large

HintThink about whether the rounding moved the number up or down before it was cubed.

The why3.9 rounds up to 4, and 4³ = 64. Because the base was rounded up, the estimate is above the exact value, which is about 59.3. Cubing makes a small rounding change noticeably bigger.

8.5, because 8.5² = 72.25, which is more than 70

HintCompare the squares of the two numbers instead of the numbers themselves.

The whyFor positive numbers, the bigger number has the bigger square. The square of √70 is 70 and the square of 8.5 is 72.25, so 8.5 is the larger. In fact √70 is about 8.37.

The power 1/2

HintAsk what number, multiplied by itself, would bring the index back up to 1.

The whyBy the index laws, a to the power ½ multiplied by itself gives a¹ = a, so a to the power ½ is the number that multiplies by itself to give a: the square root. In the same way, a to the power ⅓ is the cube root. For example, 25 to the power ½ is 5.

9

HintDeal with the bottom of the fraction first, then the top.

The whyThe denominator 3 means cube root and the numerator 2 means square. ∛27 = 3, and 3² = 9. Taking the root first keeps the numbers small: squaring first gives 729, whose cube root is harder to spot.

1/4

HintHandle the minus sign and the 1/2 as two separate jobs: one flips, the other takes a root.

The whyThe ½ means square root: √16 = 4. The minus sign means reciprocal: one over that. So 16 to the power −½ is 1/4. The two steps can be done in either order.

root

HintThink about what a power of 1/2 or 1/3 does to a number.

The whyThe denominator 3 means cube root and the numerator 2 means square, so 8 to the power ⅔ = (∛8)² = 2² = 4. In general, a to the power m/n is the nth root of a, raised to the power m.

9 to the power ½, multiplied by itself, is 9¹ = 9, and the positive number that multiplies by itself to make 9 is 3

HintSquare the quantity and see what the index laws give.

The whyWhen powers of the same base are multiplied, the indices add: ½ + ½ = 1. So 9 to the power ½ is a number whose square is 9. Fractional powers are not a new rule; they are what the index laws force.