Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Simplify expressions using the laws of exponents.
• Use scientific notation for very large and small numbers.
• Calculate with numbers in scientific notation.
• Solve problems in practical contexts.
• Identify rational and irrational numbers.

Lesson summary

This week, we delve deeper into the fascinating world of real numbers, exponents, and scientific notation. These concepts are not just abstract mathematical ideas; they are essential tools for understanding and interacting with the world around us. From calculating interest rates on loans to understanding the vast distances in space and the tiny sizes of nanoparticles, these concepts provide a framework for quantitative reasoning. Consider the population growth rate in Gauteng or the spread of a disease; understanding exponents can help us model and predict such scenarios.

Lesson notes

Real Numbers: The set of real numbers encompasses all rational and irrational numbers.

Rational Numbers: Numbers that can be expressed as a fraction p/q, where p and q are integers and q is not zero.

Examples: 1/2, -3/4, 5, 0, 0.333... (repeating decimals). Terminating decimals are also rational.

Irrational Numbers: Numbers that cannot be expressed as a fraction of two integers. Their decimal representations are non-terminating and non-repeating.

Examples: √2, π, √

3. Note that surds (roots of numbers that are not perfect squares, cubes, etc.) are often irrational.

Exponents and the Laws of Exponents: An exponent indicates how many times a base is multiplied by itself. For example, in a n , a is the base and n is the exponent. Laws of Exponents (where a and b are real numbers and m and n are integers): Product of Powers: a m a n = a m+n * (When multiplying powers with the same base, add the exponents.) Why? a m means 'a multiplied by itself m times' and a n * means 'a multiplied by itself n times'.

Therefore, when you multiply them together, you're multiplying 'a' by itself a total of m+n times.

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.) where a≠0 Why? Similar logic to the product of powers. When dividing, you are essentially cancelling out 'n' number of 'a's from both numerator and denominator, leaving you with m-n 'a's.

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.) Why? (a m ) n means you're taking a m and multiplying it by itself 'n' times. This effectively gives you a series of a m+m+m...+m (n times), which simplifies to a mn .

Example: (5 2 ) 3 = 5 23 = 5 6 = 15625 Power of a Product: (ab) n = a n b n (The power of a product is the product of the powers.) Why? (ab) n is equal to (ab)(ab)(ab)...(ab) 'n' times. Because multiplication is commutative and associative, we can rearrange this as (aaa...a)(bb*b...b), where both 'a' and 'b' are multiplied by themselves 'n' times.

Example: (2x) 3 = 2 3 x 3 = 8x 3 Power of a Quotient: (a/b) n = a n / b n (The power of a quotient is the quotient of the powers.) where b≠0 Why? Similar to the product rule, (a/b) n means (a/b)(a/b)(a/b)...(a/b) 'n' times. This equals (aaa...a) / (bb*b...b), which simplifies to a n / b n .

Example: (4/y) 2 = 4 2 / y 2 = 16/y 2 Zero Exponent: a 0 = 1 (Any non-zero number raised to the power of zero is equal to 1.) where a≠0 Why? This is a consequence of the quotient rule. Consider a n / a n . We know this equals

1. However, using the quotient rule, it also equals a n-n = a 0 .

Therefore, a 0 * must equal

1. Example: 7 0 = 1 Negative Exponent: a -n = 1 / a n (A number raised to a negative exponent is equal to the reciprocal of that number raised to the positive exponent.) where a≠0 Why? Again, this arises from the quotient rule. Consider a 0 / a n . We know a 0 = 1, so this equals 1 / a n . Using the quotient rule, this also equals a 0-n = a -n .

Therefore, a -n * must equal 1 / a n .

Example: 2 -3 = 1 / 2 3 = 1/8 Scientific Notation: A way of writing very large or very small numbers in the form a x 10 n , where 1 ≤ |a| 6 Small Numbers: Move the decimal point to the right until there is only one non-zero digit to the left of the decimal point. The number of places moved becomes the negative exponent n.

Example: 0.000056 = 5.6 x 10 -5 Calculations with Scientific Notation: Multiplication: Multiply the 'a' values and add the exponents. (a x 10 m ) (b x 10 n ) = (a*b) x 10 m+n Division: Divide the 'a' values and subtract the exponents. (a x 10 m ) / (b x 10 n ) = (a/b) x 10 m-n Addition/Subtraction: The numbers must have the same exponent for

1

0. If not, adjust one of the numbers. Then add/subtract the 'a' values, keeping the exponent for 10 the same. Convert both numbers to the same power of 10: a x 10 n + b x 10 n = (a+b) x 10 n * Guided Practice (With Solutions)

Question 1: Simplify: (3x 2 y -1 ) 3 Solution: Apply the power of a product rule: (3x 2 y -1 ) 3 = 3 3 (x 2 ) 3 (y -1 ) 3 Simplify each term using the power of a power rule: 27x 6 y -3 Rewrite with positive exponents: 27x 6 / y 3

Commentary: This question combines the power of a product and power of a power rules. Remember to apply the exponent to every term inside the parentheses. The final step ensures the answer is written with positive exponents, which is often preferred.

Question 2: Express 0.000478 in scientific notation.

Solution: Move the decimal point 4 places to the right until there's one non-zero digit to the left of the decimal: 4.78 Since we moved the decimal 4 places to the right, the exponent is -

4. Therefore, 0.000478 = 4.78 x 10 -4

Commentary: The direction you move the decimal is crucial. Moving right gives a negative exponent, and moving left gives a positive exponent. Always double-check that the absolute value of the number before the power of 10 is between 1 and 10.