Powers, roots and index laws
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교육자를 위해: Powers, roots and index laws(KS3 Maths, Number)을(를) 위한 바로 쓸 수 있는 수업 슬라이드, 복습 노트 — 수업에 사용하거나, 학습자들이 실시간 게임으로 즐기는 인터랙티브 클래스 활동으로 진행하세요.
수업 노트
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.
슬라이드
연습 문제
무료 미리 보기 — 64개 중 8개 문제. 가입하면 전부 볼 수 있어요.
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
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