Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Classify numbers as rational or irrational.
• Simplify expressions using the laws of exponents.
• Express numbers in scientific notation and perform calculations.
• Apply these concepts to solve real-world problems.
• Evaluate expressions involving exponents and roots.

Lesson summary

This week, we delve into the fascinating world of real numbers, exponents, and scientific notation. These concepts are fundamental to understanding mathematics and have far-reaching applications in various fields, from science and engineering to finance and everyday problem-solving. In South Africa, understanding these concepts can help you interpret economic data, understand scientific reports on climate change, and even calculate interest rates on loans or investments. Imagine understanding how much faster a new train will get you to a relative in another province, or understanding how data from the latest Census represents our population!

Lesson notes

2.1 Real Numbers: Real numbers encompass all rational and irrational numbers.

Rational Numbers: These are numbers that can be expressed as a fraction p/q, where p and q are integers and q ≠

0. Examples: 2/3, -5, 0.75 (which is 3/4). Terminating and repeating decimals are rational.

Irrational Numbers: These are numbers that cannot be expressed as a fraction p/q. Their decimal representations are non-terminating and non-repeating.

Examples: √2, π (pi), √

5. Example 1: Classify the following numbers as rational or irrational: 3, √16, π, 0.333..., √7, -2/5 3: Rational (3/1) √16: Rational (√16 = 4 = 4/1) π: Irrational 0.333...: Rational (repeating decimal, equivalent to 1/3) √7: Irrational -2/5: Rational 2.2 Exponents (Indices): An exponent indicates how many times a base number is multiplied by itself. In the expression a n , 'a' is the base and 'n' is the exponent.

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 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 Power of a Power: (a m ) n = a mn (When raising a power to another power, multiply the exponents)

Example: (5 2 ) 3 = 5 23 = 5 6 = 15625 Zero Exponent: a 0 = 1 (Any non-zero number raised to the power of 0 is 1)

Example: 7 0 = 1 Negative Exponents: a -n = 1/a n (A negative exponent indicates the reciprocal of the base raised to the positive exponent)

Example: 4 -2 = 1/4 2 = 1/16 Example 2: Simplify the following expressions: a) x 4 * x 7 b) y 9 / y 3 c) (z 2 ) 5 d) 6 0 e) 5 -3 a) x 4 * x 7 = x 4+7 = x 11 b) y 9 / y 3 = y 9-3 = y 6 c) (z 2 ) 5 = z 2*5 = z 10 d) 6 0 = 1 e) 5 -3 = 1/5 3 = 1/125 2.3 Scientific Notation: Scientific notation is a way of expressing very large or very small numbers in a compact and convenient form. It is written as a × 10 n , where 1 ≤ |a| 6 (Move the decimal point 6 places to the left) b) 0.000052 = 5.2 x 10 -5 (Move the decimal point 5 places to the right)

Example 4: Express the following numbers in standard form: a) 6.7 x 10 4 b) 9.1 x 10 -3 a) 6.7 x 10 4 = 67,000 (Move the decimal point 4 places to the right) b) 9.1 x 10 -3 = 0.0091 (Move the decimal point 3 places to the left)

Example 5: Calculations with Scientific Notation Simplify (2 x 10 3 ) x (3 x 10 4 ) (2 x 10 3 ) x (3 x 10 4 ) = (2 x 3) x (10 3 x 10 4 ) = 6 x 10 7 Guided Practice (With Solutions)

Question 1: Simplify: (4 2 * 4 -1 ) / 4 0 Solution: (4 2 * 4 -1 ) / 4 0 = 4 2+(-1) / 4 0 = 4 1 / 4 0 = 4 1 / 1 =

4. Commentary: We first applied the product of powers rule in the numerator, then simplified 4 0 to 1, and finally performed the division.

Question 2: Express 0.000785 in scientific notation.

Solution: 000785 = 7.85 x 10 -4

Commentary: We moved the decimal point four places to the right to get a number between 1 and 10, and since we moved it to the right, the exponent is negative.

Question 3: Classify √25, 1/7, and √10 as rational or irrational.

Solution: √25 = 5, which is rational. 1/7 is rational (it's a fraction). √10 is irrational (it doesn't have a perfect square root and its decimal representation is non-terminating and non-repeating).

Commentary: Remember that a perfect square yields a rational number when square rooted.

Question 4: Simplify (5 x 10 -2 ) / (2.5 x 10 -5 ) and express the answer in scientific notation.

Solution: (5 x 10 -2 ) / (2.5 x 10 -5 ) = (5/2.5) x (10 -2 / 10 -5 ) = 2 x 10 (-2 - (-5)) = 2 x 10 3

Commentary: We divided the coefficients and applied the quotient rule to the powers of

1

0. Independent Practice (Questions Only)

Simplify: a 5 a -2 a 0 Simplify: (b 3 ) 4 / b 6 Express 0.000000314 in scientific notation. Express 5.89 x 10 7 in standard form.

Simplify: (3 x 10 5 ) + (7 x 10 4 ) (Hint: convert to same power of 10 first). Classify the following numbers as rational or irrational: 0.666..., √11, -8, 3/4 Evaluate (√4) 3 Simplify: 8 -2 * 2 5 The population of Gauteng province is estimated to be 16 million. Express this number in scientific notation. The approximate size of a virus is 0.00000002 meters. Express this in scientific notation.