Grade 7
  1. Number System  1.1. Integers  1.1.1. Properties of Integers  1.1.2. Operations on Integers  1.1.3. Additive and Multiplicative Inverses  1.1.4. Applications of Integers  1.2. Rational Numbers  1.2.1. Definition of Rational Numbers  1.2.2. Properties of Rational Numbers  1.2.3. Operations on Rational Numbers  1.2.4. Standard Form of Rational Numbers  1.3. Decimals  1.3.1. Operations on Decimals  1.3.2. Converting Between Fractions and Decimals  1.4. Powers and Exponents  1.4.1. Laws of Exponents  1.4.2. Simplifying Expressions with Exponents  1.4.3. Scientific Notation  1.5. Roots  1.5.1. Square Roots  1.5.2. Cube Roots
  2. Algebra  2.1. Expressions  2.1.1. Algebraic Expressions  2.1.2. Simplifying Expressions  2.1.3. Evaluating Expressions  2.2. Linear Equations  2.2.1. Solving Simple Equations  2.2.2. Applications of Linear Equations  2.3. Algebraic Identities  2.3.1. Expanding Using Identities  2.3.2. Simplifying Using Identities
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Simplifying Ratios  3.1.3. Equivalent Ratios  3.2. Proportion  3.2.1. Concept of Proportion  3.2.2. Direct Proportion  3.2.3. Inverse Proportion  3.3. Unitary Method  3.3.1. Solving Problems Using the Unitary Method
  4. Geometry  4.1. Lines and Angles  4.1.1. Types of Angles  4.1.2. Pairs of Angles  4.1.3. Properties of Parallel Lines and a Transversal  4.1.4. Angle Bisectors  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Angle Sum Property  4.2.3. Exterior Angle Property  4.2.4. Pythagoras Theorem  4.3. Congruence of Triangles  4.3.1. Criteria for Congruence  4.3.2. Applications of Congruence  4.4. Quadrilaterals  4.4.1. Types of Quadrilaterals  4.4.2. Properties of Parallelograms  4.4.3. Special Quadrilaterals  4.4.4. Properties of Rectangles and Rhombuses
  5. Mensuration  5.1. Perimeter and Area  5.1.1. Perimeter of Plane Figures  5.1.2. Area of Plane Figures  5.1.3. Area of a Trapezium  5.1.4. Area of a Parallelogram  5.1.5. Area of a Circle  5.1.6. Area of Composite Figures  5.2. Surface Area and Volume  5.2.1. Cubes and Cuboids  5.2.2. Surface Area of Cylinders  5.2.3. Volume of Prisms and Cylinders  5.2.4. Volume and Surface Area of Cones
  6. Data Handling  6.1. Data Collection  6.1.1. Organizing Data  6.1.2. Frequency Distribution  6.2. Graphical Representation of Data  6.2.1. Bar Graphs  6.2.2. Double Bar Graphs  6.2.3. Pie Charts  6.2.4. Histograms  6.2.5. Line Graphs  6.3. Probability  6.3.1. Simple Events  6.3.2. Experimental Probability  6.3.3. Theoretical Probability
  7. Practical Geometry  7.1. Construction of Triangles  7.1.1. Constructing Triangles Given Side Lengths  7.1.2. Constructing Triangles Given Side and Angle  7.1.3. Constructing Right-Angled Triangles  7.2. Construction of Quadrilaterals  7.2.1. Quadrilaterals Given Side Lengths and Angles  7.2.2. Constructing Parallelograms and Rectangles

Grade 7 → Number System → Powers and Exponents ↓


Laws of Exponents


Exponents are used to show how many times a number is multiplied by itself. For example, in the expression 2^3, the number 2 is multiplied by itself 3 times: 2 * 2 * 2 is important to understand the rules of exponents because they help simplify expressions and solve problems involving exponents.

What are exponents?

The exponent is the number that tells how many times a base number is to be multiplied by itself. In the expression a^n, a is the base and n is the exponent. The expression means that a is multiplied by itself n times.

Example:

3^4 = 3 * 3 * 3 * 3 = 81

Basic laws of exponents

Let's take a look at some basic rules of exponents. These rules are fundamental to simplifying expressions and solving equations.

1. Product rule of powers

When multiplying two powers with the same base, keep the base and add the exponents. In other words:

a^m * a^n = a^(m+n)

Example:

2^3 * 2^4 = 2^(3+4) = 2^7 = 2 * 2 * 2 * 2 * 2 * 2 * 2 = 128

2. Power quotient rule

When dividing two powers with the same base, keep the base and subtract the exponents.

a^m / a^n = a^(m-n)

Example:

5^5 / 5^3 = 5^(5-3) = 5^2 = 5 * 5 = 25

3. The power of the power rule

When raising a power to another power, keep the base and multiply the exponents.

(a^m)^n = a^(m*n)

Example:

(3^2)^3 = 3^(2*3) = 3^6 = 3 * 3 * 3 * 3 * 3 * 3 = 729

4. The power of the multiplication rule

To find the power of a product, distribute the exponent of each factor inside the parentheses.

(ab)^n = a^n * b^n

Example:

(2 * 3)^2 = 2^2 * 3^2 = 4 * 9 = 36

5. Rule of power of quotient

Apply the exponent to both the numerator and denominator to find the power of the quotient.

(a/b)^n = a^n / b^n

Example:

(4/2)^3 = 4^3 / 2^3 = 64 / 8 = 8

6. Zero exponent rule

Any base (except 0) raised to the zero power is equal to 1.

a^0 = 1

Example:

7^0 = 1 100^0 = 1

7. Negative exponent rule

Negative exponents mean how many times the number has to be divided. This is the opposite of multiplying. In other words:

a^(-n) = 1/a^n

Example:

2^(-3) = 1/(2^3) = 1/8

Visual example

Here are visual examples using the rules of numbers and exponents:

Example of product of powers:

2^2 * 2^3 = 2^(2+3) = 2^5 Diagram: 2^2 = 2 * 2 = 4 2^3 = 2 * 2 * 2 = 8 Multiply: 4 * 8 = 2^5 Calculations confirm: 2^5 = 32

Example of a quotient of powers:

10^4 / 10^2 = 10^(4-2) = 10^2 Diagram: 10^4 = 10 * 10 * 10 * 10 = 10000 10^2 = 10 * 10 = 100 Divide: 10000 / 100 = 10^2 Calculations confirm: 10^2 = 100

Example of the power of power:

(5^2)^3 = 5^(2*3) = 5^6 Diagram: 5^2 = 5 * 5 = 25 25 raised to the power 3 (25^3) Calculations confirm: 5^6 = 15625

Example of the power of a product:

(3 * 2)^2 = 3^2 * 2^2 Diagram: 3 * 2 = 6 6 to the power of 2 = 36 Different calculations: 3^2 = 9, 2^2 = 4, and 9 * 4 = 36

Additional examples and problems

Here are some additional examples to practice the rules of exponents and understand them more deeply:

Practice problem 1:

Simplify 4^7 * 4^2.

Solution: Using the product rule of powers: 4^7 * 4^2 = 4^(7+2) = 4^9 = 262144

Practice problem 2:

Simplify 9^5 / 9^3.

Solution: Using the power quotient rule: 9^5 / 9^3 = 9^(5-3) = 9^2 = 81

Practice problem 3:

Simplify (6^2)^4.

Solution: Using the power of powers rule: (6^2)^4 = 6^(2*4) = 6^8 = 1679616

Practice problem 4:

Simplify (4 * 7)^2.

Solution: Using the power of product rule: (4 * 7)^2 = 4^2 * 7^2 = 16 * 49 = 784

Practice problem 5:

Simplify (5/3)^3.

Solution: Using the quotient power rule: (5/3)^3 = 5^3 / 3^3 = 125 / 27

Understanding zero and negative exponents

In addition to operations with positive exponents, it is important to understand how zero and negative exponents work.

Zero exponent example:

7^0 = 1 Explanation: Any non-zero number raised to the power of 0 gives 1. This rule arises from the pattern observed in dividing exponential numbers: for a^n/a^n = a^(n-n) = a^0 = 1.

Negative exponent example:

3^(-2) = 1/(3^2) Explanation: Negative exponents indicate the inverse. If the exponent of the base is negative, take the reciprocal of the base and apply the positive exponent. In calculation, 3^(-2) = 1/9.

Conclusion

Understanding the rules of exponents simplifies many mathematical processes. By remembering these rules, you can solve equations efficiently, whether working with algebraic expressions or working on practical calculations in various fields. Practice these concepts and rules to become proficient at handling exponents and exponents.


Grade 7 → 1.4.1


U
username
0%
completed in Grade 7

← Prev (1.4)
Powers and Exponents
Next (1.4.2) →
Simplifying Expressions with Exponents

Comments 



Laws of Exponents

https://www.buddymath.com/g7/1.4.1?bmal=en