Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Simplify expressions using the laws of exponents.
• Express numbers in scientific notation and convert back.
• Perform calculations with numbers in scientific notation.
• Solve practical problems involving these concepts.

Lesson summary

This week, we delve deeper into the world of Real Numbers, building on your previous knowledge. We'll be specifically focusing on exponents and scientific notation. Exponents provide a shorthand way of writing repeated multiplication, which is crucial for understanding many mathematical and scientific concepts. Scientific notation allows us to express very large and very small numbers in a compact and manageable form. Think about South Africa's economy, population, or even the size of a virus – these often involve numbers that are either incredibly large or unbelievably small, and scientific notation is the perfect tool to handle them.

Lesson notes

2.1 Real Numbers: Recall that the real numbers include all rational and irrational numbers. Rational numbers can be expressed as a fraction p/q, where p and q are integers and q ≠

0. Irrational numbers cannot be expressed as a simple fraction (e.g., √2, π). Understanding this foundation is important as we will be manipulating real numbers using exponents. 2.2 Exponents (Powers): An exponent indicates how many times a base number is multiplied by itself. For example, in the expression a n , a is the base and n is the exponent. a n means a multiplied by itself n times: a × a × a × ... × a (n times). 2.3 Laws of Exponents: These laws are fundamental for simplifying expressions involving exponents.

Product of Powers: a m × a n = a m+n (When multiplying powers with the same base, add the exponents.)

Example:* 2 3 × 2 2 = 2 3+2 = 2 5 = 32 Why?* 2 3 × 2 2 = (2 × 2 × 2) × (2 × 2) = 2 × 2 × 2 × 2 × 2 = 2 5 Quotient of Powers: a m / a n = a m-n (When dividing powers with the same base, subtract the exponents.)

Example:* 3 5 / 3 2 = 3 5-2 = 3 3 = 27 Why?* 3 5 / 3 2 = (3 × 3 × 3 × 3 × 3) / (3 × 3) = 3 × 3 × 3 = 3 3 Power of a Power: (a m ) n = a m×n (When raising a power to another power, multiply the exponents.)

Example:* (5 2 ) 3 = 5 2×3 = 5 6 = 15625 Why?* (5 2 ) 3 = (5 2 ) × (5 2 ) × (5 2 ) = (5 × 5) × (5 × 5) × (5 × 5) = 5 × 5 × 5 × 5 × 5 × 5 = 5 6 Power of a Product: (a × b) n = a n × b n (The power of a product is the product of the powers.)

Example:* (2 × 3) 2 = 2 2 × 3 2 = 4 × 9 = 36 Why?* (2 × 3) 2 = (2 × 3) × (2 × 3) = 2 × 2 × 3 × 3 = 2 2 × 3 2 Power of a Quotient: (a / b) n = a n / b n (The power of a quotient is the quotient of the powers.)

Example:* (4 / 2) 3 = 4 3 / 2 3 = 64 / 8 = 8 Why?* (4 / 2) 3 = (4 / 2) × (4 / 2) × (4 / 2) = (4 × 4 × 4) / (2 × 2 × 2) = 4 3 / 2 3 Zero Exponent: a 0 = 1 (Any non-zero number raised to the power of 0 is 1.)

Example:* 7 0 = 1 Why? Consider a m / a m =

1. Using the quotient rule, this is also equal to a m-m = a 0 .

Therefore, a 0 =

1. Negative Exponent: a -n = 1 / a n (A number raised to a negative exponent is equal to the reciprocal of the number raised to the positive exponent.)

Example:* 2 -3 = 1 / 2 3 = 1 / 8 Why? Consider a 0 / a n = 1 / a n . Using the quotient rule, this is also equal to a 0-n = a -n .

Therefore, a -n = 1 / a n . 2.4 Scientific Notation: Scientific notation is a way of expressing numbers as a product of a number between 1 and 10 (inclusive of 1 but exclusive of 10) and a power of

1

0. The general form is a × 10 n , where 1 ≤ |a| 5 Example 2:* Convert 0.000027 to scientific notation. Move the decimal point 5 places to the right: 2.7 Therefore, 0.000027 = 2.7 × 10 -5 Converting from Scientific Notation: Move the decimal point the number of places indicated by the exponent of

1

0. If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left.

Example 1:* Convert 4.8 × 10 3 to ordinary notation. Move the decimal point 3 places to the right: 4800 Therefore, 4.8 × 10 3 = 4800 Example 2:* Convert 9.1 × 10 -4 to ordinary notation. Move the decimal point 4 places to the left: 0.00091 Therefore, 9.1 × 10 -4 = 0.00091 2.5 Operations with Scientific Notation: Multiplication: Multiply the decimal numbers and add the exponents of 10. (a × 10 m ) × (b × 10 n ) = (a × b) × 10 m+n *

Example:* (2 × 10 3 ) × (3 × 10 4 ) = (2 × 3) × 10 3+4 = 6 × 10 7 Division: Divide the decimal numbers and subtract the exponents of 10. (a × 10 m ) / (b × 10 n ) = (a / b) × 10 m-n *

Example:* (8 × 10 5 ) / (2 × 10 2 ) = (8 / 2) × 10 5-2 = 4 × 10 3 Addition and Subtraction: The exponents of 10 must be the same before adding or subtracting. Adjust one of the numbers to have the same exponent as the other.

Example:* (3 × 10 4 ) + (2 × 10 3 ) = (3 × 10 4 ) + (0.2 × 10 4 ) = (3 + 0.2) × 10 4 = 3.2 × 10 4 Guided Practice (With Solutions)

Question 1: Simplify: (3 2 × 3 4 ) / 3 3 Solution: Step 1: Apply the product of powers rule to the numerator: 3 2 × 3 4 = 3 2+4 = 3 6 Step 2: Apply the quotient of powers rule: 3 6 / 3 3 = 3 6-3 = 3 3 Step 3: Evaluate: 3 3 = 3 × 3 × 3 = 27 Answer: 27

Commentary: This question combines two laws of exponents. Make sure to follow the order of operations.

Question 2: Express 0.00000519 in scientific notation.

Solution: Step 1: Move the decimal point 6 places to the right to get a number between 1 and 10: 5.19 Step 2: Since we moved the decimal point 6 places to the right, the exponent of 10 will be -

6. Answer: 5.19 × 10 -6

Commentary: Be careful with the sign of the exponent. Moving to the right makes the exponent negative.

Question 3: Simplify: (4 × 10 5 ) × (2.5 × 10 -2 )

Solution: Step 1: Multiply the decimal numbers: 4 × 2.5 = 10 Step 2: Add the exponents of 10: 10 5 × 10 -2 = 10 5+(-2) = 10 3 Step 3: Combine the results: 10 × 10 3 = 1 × 10 1 × 10 3 = 1 × 10 4 Answer: 1 × 10 4 or 10000

Commentary: The result can be simplified further since 10 is already a power of 10.