Powers, roots and index laws
遊んで学ぼう
問題に答えてエネルギーを集めたら、釣りや探検を楽しもう。アカウント不要。
教育者の方へ： Powers, roots and index laws（KS3 Maths、Number）向けのすぐ使えるレッスンスライド, 復習ノート — レッスンで使うか、学習者がライブゲームとして遊ぶインタラクティブなクラス活動としてトピックを実施できます。
レッスンノート
Squares, Cubes and Higher Powers
- A power is a number multiplied by itself a certain number of times.
- The base is the number being multiplied; the exponent (or index) tells you how many times to multiply it.
- For example, bn means b imes b imes \dots imes b with n copies of b.
- b2 is read as 'b squared' because it gives the area of a square with side length b.
- b3 is read as 'b cubed' because it gives the volume of a cube with side length b.
- Higher powers like b4, b5 follow the same pattern: multiply the base by itself that many times.
Index Notation
- We write powers using a superscript: bn, where b is the base and n is the index.
- The index is also called the exponent or power.
- You may see the caret symbol used instead: bn.
- The expression bn is read as 'b to the power of n' or 'b to the n'.
- The result of a power calculation is also called a power.
Square Roots and Cube Roots
- The square root of a number is the value that, when squared, gives that number.
- The square root of x is written \√{x}.
- For example, \√{25} = 5 because 52 = 25.
- The cube root of a number is the value that, when cubed, gives that number.
- The cube root of x is written \√[3]{x}.
- For example, \√[3]{27} = 3 because 33 = 27.
- Roots can be estimated: \√{50} is between 7 and 8 because 72 = 49 and 82 = 64.
Multiplying Powers with the Same Base
- When multiplying powers with the same base, add the indices: am imes an = am+n.
- For example, 23 imes 24 = 23+4 = 27.
- This works because you are combining the repeated multiplications.
- The base stays the same; only the index changes.
Dividing Powers with the Same Base
- When dividing powers with the same base, subtract the indices: am \div an = am-n.
- For example, 56 \div 52 = 56-2 = 54.
- Again, the base remains the same.
- This rule works when the first index is larger than the second.
Powers of a Power
- To raise a power to another power, multiply the indices: (am)n = am imes n.
- For example, (32)4 = 32 imes 4 = 38.
- This is because you are multiplying the base by itself m imes n times in total.
The Zero Index
- Any non-zero number raised to the power 0 equals 1: a0 = 1.
- For example, 70 = 1 and (-3)0 = 1.
- This follows from the division rule: an \div an = an-n = a0, and any number divided by itself is 1.
Negative Indices
- A negative index means the reciprocal of the positive power: a-n = rac{1}{an}.
- For example, 2-3 = rac{1}{23} = rac{1}{8}.
- Also, a-1 = rac{1}{a}.
- Negative indices do not make the result negative; they make it a fraction.
Powers of 10
- Powers of 10 are easy to calculate: 10n is a 1 followed by n zeros.
- For example, 103 = 1000.
- Negative powers of 10 give decimals: 10-1 = 0.1, 10-2 = 0.01, 10-3 = 0.001.
- Powers of 10 are useful for writing very large or very small numbers in standard form.
スライド
練習問題
無料プレビュー — 64問中8問。すべて見るには登録を。
1.In the expression bn, what is the number n called?
Easy- Athe base
- Bthe exponent
- Cthe coefficient
- Dthe product
2.In the expression bn, the number b is called the base.
EasyTrue or false?
3.Which of the following are correct names for the exponent in a power? (select all that apply)
Easy- Aindex
- Bpower
- Cbase
- Dcoefficient
- Eroot
4.Which expression is equivalent to 5 × 5 × 5 × 5?
Easy- A53
- B54
- C45
- D5 × 4
5.What is the value of 34?
Medium- A12
- B64
- C81
- D27
6.What is the value of 60?
Medium- A0
- B1
- C6
- D60
7.Simplify 75 × 73, leaving your answer as a power of 7.
Medium- A78
- B715
- C72
- D498
8.Simplify 89 ÷ 84, leaving your answer as a power of 8.
Medium- A813
- B85
- C836
- D15
過去問
このトピックの過去問練習は近日公開。
近日公開