Powers, roots and index laws

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课程笔记

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: a^m imes aⁿ = a^m+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: a^m \div aⁿ = a^m-n.
For example, 5⁶ \div 5² = 5^6-2 = 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: (a^m)ⁿ = a^{m imes n}.
For example, (3²)⁴ = 3^{2 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 = a^n-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.

幻灯片

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练习题

免费预览——64 题中的 8 题。注册即可查看全部。

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.
Easy
True 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

复习笔记

## Powers, Roots and Index Laws **Index notation** A power is written b^n: b is the base, n is the exponent (also called the power or the index). b^n = b × b × ... × b (n times). - b^1 = b - b^2 is "b squared" (area of a square of side b) - b^3 is "b cubed" (volume of a cube of side b) **Squares, cubes and roots** - Square root: √(b^2) = b. The square root of a square number is a whole number. - Cube root: the number that multiplies by itself three times to give the cube. - Estimate roots by finding the two square (or cube) numbers the value lies between. **Laws of indices** (same base, a ≠ 0) - Multiplying: a^m × a^n = a^{m+n} - Dividing: a^m ÷ a^n = a^{m−n} - Power of a power: (a^m)^n = a^{mn} - Zero index: a^0 = 1 - Negative index: a^{−n} = 1 / a^n **Powers of 10** 10^1 = 10, 10^2 = 100, 10^3 = 1000, ... 10^0 = 1, 10^{−1} = 0.1, 10^{−2} = 0.01. The index of 10 tells you how many zeros (positive index) or how many decimal places (negative index). **Quick checks** - 2^3 × 2^4 = 2^7 = 128 - 5^6 ÷ 5^2 = 5^4 = 625 - (3^2)^3 = 3^6 = 729 - 7^0 = 1 - 4^{−1} = 1/4 = 0.25

Base: The number being multiplied repeatedly in a power, written b in b^n.
Exponent (index, power): The number that tells how many times the base is multiplied by itself, written n in b^n.
Power: The result of exponentiation, or another name for the exponent; b^n is read as 'b to the power of n'.
Square: b^2 = b × b, called 'b squared' because it gives the area of a square of side b.
Cube: b^3 = b × b × b, called 'b cubed' because it gives the volume of a cube of side b.
Square root: The number that multiplies by itself to give the square; √(b^2) = b.
Cube root: The number that multiplies by itself three times to give the cube.
Index law for multiplying: a^m × a^n = a^{m+n} (same base).
Index law for dividing: a^m ÷ a^n = a^{m−n} (same base).
Power of a power: (a^m)^n = a^{mn}.
Zero index: a^0 = 1 for any non-zero base a.
Negative index: a^{−n} = 1 / a^n.

闪卡

What is the base and what is the exponent in 5³?

The base is 5 and the exponent is 3. The exponent is also called the power or index.

What does 5³ mean as repeated multiplication?

5³ = 5 × 5 × 5. In general, bⁿ is b multiplied by itself n times.

What is the value of 5³?

5³ = 125.

What is the square of 7?

The square of 7 is 7² = 49.

What is the cube of 4?

The cube of 4 is 4³ = 64.

What is the square root of 81?

The square root of 81 is 9, because 9² = 81.

What is the cube root of 27?

The cube root of 27 is 3, because 3³ = 27.

Estimate the square root of 50 to the nearest whole number.

7² = 49 and 8² = 64, so √50 is about 7.

State the index law for multiplying powers with the same base.

a^m × aⁿ = a^m+n. For example, 2³ × 2⁴ = 2⁷.

State the index law for dividing powers with the same base.

a^m ÷ aⁿ = a^m-n. For example, 5⁶ ÷ 5² = 5⁴.

State the index law for a power of a power.

(a^m)ⁿ = a^m×n. For example, (3²)⁴ = 3⁸.

What is the value of any non-zero number raised to the power 0?

a⁰ = 1 for any non-zero a. For example, 7⁰ = 1.

What does a negative index mean?

a^-n = 1 / aⁿ. For example, 2^-3 = 1 / 2³ = 1/8.

Write 10^-3 as a fraction.

10^-3 = 1 / 10³ = 1/1000.

What is 10⁴?

10⁴ = 10 × 10 × 10 × 10 = 10000.

Simplify 3⁵ × 3².

3⁵ × 3² = 3⁵⁺² = 3⁷ = 2187.

Simplify 4⁷ ÷ 4³.

4⁷ ÷ 4³ = 4^7-3 = 4⁴ = 256.

Simplify (2³)⁵.

(2³)⁵ = 2^3×5 = 2¹⁵ = 32768.

What is the value of 6^-1?

6^-1 = 1 / 6¹ = 1/6.

What is the value of 9^1/2?

9^1/2 = √9 = 3.

What is the value of 8^1/3?

8^1/3 = ∛8 = 2.

Write 2⁵ in expanded form.

2⁵ = 2 × 2 × 2 × 2 × 2.

What is the value of 2⁵?

2⁵ = 32.

What is the value of 3⁴?

3⁴ = 81.

What is the value of 10⁰?

10⁰ = 1.

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