Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Exponents and Powers ↓


Laws of Exponents


The rules of exponents are rules that explain how to handle mathematical operations involving powers of numbers. Exponents are used to represent numbers that are multiplied by themselves a certain number of times. In this lesson, we will explore these rules in detail, and provide plenty of examples to help you understand how they work in different situations. Let's begin our journey into the wonderful world of exponents!

Understanding exponents

Before we dive deeper into the rules, it's important to understand what exponents are. The exponent is a small number written in the upper-right corner of the base number, indicating how many times the base number has been multiplied by itself.

For example, in the expression 5 3 5 is the base, and 3 is the exponent. This expression means multiplying the base number 5 by itself 3 times:

5 × 5 × 5 = 125

Exponents make large or repeated multiplications easier to write and handle.

Laws of exponents

These rules are powerful tools in simplifying expressions and solving equations involving exponents. Let's explore each rule one by one.

1. Product of powers

The product rule of powers states that when you multiply two numbers with the same base, you add their exponents. Mathematically, this is represented as:

am × an = am+n

Let's look at an example:

2 3 × 2 4 = 2 3+4 = 2 7

Its analysis:

2 × 2 × 2 × 2 × 2 × 2 × 2 = 2 × 2 × 2 × 2 × 2 × 2 × 2

Note that the base remains the same (2), and we add the exponents: 3 + 4.

2 3 × 2 4 = 2 7

2. Quotient of powers

The power quotient rule states that when you divide two numbers with the same base, you subtract the exponent of the denominator from the exponent of the numerator:

am ÷ an = am-n

Here's an example:

5 4 ÷ 5 2 = 5 4-2 = 5 2

Breakdown:

5 × 5 × 5 × 5 ÷ (5 × 5) = 5 × 5

As you can see, the base (5) remains unchanged, and we have reduced the exponent 4 - 2 to simplify the expression.

3. The power of power

According to the power rule, when raising a power to another power, you multiply the exponents:

(am)n = am × n

Example:

(3 2) 3 = 3 2×3 = 3 6

Respectively:

(3 × 3)3 = 3 × 3 × 3 × 3 × 3 × 3
(am)n = am × n

4. Power of the product

The power of the product rule means that when you raise a product to an exponent, it applies to each of the factors:

(a × b)n = an × bn

Take this example:

(2 × 3) 2 = 2 2 × 3 2

The solution to this:

(2 × 3) × (2 × 3) = (2 × 2) × (3 × 3) = 4 × 9 = 36

5. Power of the quotient

According to the quotient power rule, if the quotient is raised to an exponent, then the exponent applies to both the numerator and the denominator:

(a ÷ b)n = an ÷ bn

Example:

(4 ÷ 2) 3 = 4 3 ÷ 2 3

Description:

(4 ÷ 2) × (4 ÷ 2) × (4 ÷ 2) = (4 × 4 × 4) ÷ (2 × 2 × 2) = 64 ÷ 8 = 8

6. Zero exponent

The zero exponent rule is interesting because, according to this rule, the power of a nonzero base is zero 1:

a0 = 1 
(where a ≠ 0)

Example:

6 0 = 1

Another example:

1000 0 = 1

This rule emphasizes that no matter how complex or large a value is, it will always simplify to 1 when raised to the power of zero.

7. Negative exponent

The negative exponent rule involves inverses. The rule says that a negative exponent means you take the inverse of the base:

a-n = 1/an

Example:

2 -3 = 1/2 3 = 1/8

Another example:

5 -1 = 1/5

This rule is helpful in simplifying expressions and converting them into their inverse forms.

Simplification of expressions

Now that we have discussed each rule, let's see how we can use them to simplify complex expressions. Below are some examples:

Simplify: 2 3 × 2 -1 ÷ 2 2

= 23 + (-1) - 2
= 20
= 1

Simplify: (3 2 × 4 2) 1/2

= (32)1/2 × (42)1/2
= 3 × 4
= 12

Conclusion

The rules of exponents are fundamental in math, providing the tools necessary to simplify expressions, solve equations, and understand exponential growth and decay. Whether multiplying powers, dividing them, or working with negative and zero exponents, these rules keep us grounded and provide clarity. With practice, you'll find that these rules help you solve a wide range of mathematical problems more efficiently. Keep experimenting with different expressions to solidify your understanding and increase your confidence in using these rules.


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Laws of Exponents

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