Number, Year 10: counting, powers and roots

24

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

为什么There 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

提示Its name uses the word for the answer to a multiplication.

为什么With 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

提示Once the first letter is used it cannot be picked again.

为什么Any 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

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

为什么Each 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

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

为什么Starter 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

提示Find the square numbers on either side of 50.

为什么7² = 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

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

为什么90 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

提示List the first few cube numbers until you pass 100.

为什么4³ = 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

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

为什么3.9 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

提示Compare the squares of the two numbers instead of the numbers themselves.

为什么For 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

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

为什么By 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

提示Deal with the bottom of the fraction first, then the top.

为什么The 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

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

为什么The 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

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

为什么The 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

提示Square the quantity and see what the index laws give.

为什么When 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.