Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Apply the laws of exponents to simplify expressions.
• Simplify surds and perform basic arithmetic operations.
• Rationalise denominators containing surds.
• Solve simple exponential equations.
• Understand the relationship between exponents and radicals.

Lesson summary

Exponents and surds are fundamental building blocks in mathematics. Understanding them is crucial, not just for future math topics like calculus and trigonometry, but also for applications in everyday life. For example, understanding exponential growth helps in comprehending population increases, compound interest on savings, or the spread of information on social media – all highly relevant in a South African context. Surds, while seemingly abstract, are essential for precise calculations in fields like engineering, architecture, and even certain aspects of finance.

Lesson notes

2. 1.

Exponents and Exponential Laws: 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. Understanding and applying the laws of exponents is critical for simplifying complex expressions.

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

Quotient of Powers: a m / a n = a m-n * (When dividing powers with the same base, subtract the exponents)

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

Power of a Product: (ab) n = a n b n * (A product raised to a power is equal to each factor raised to that power)

Power of a Quotient: (a/b) n = a n /b n * (A quotient raised to a power is equal to the numerator raised to that power divided by the denominator raised to that power)

Zero Exponent: a 0 = 1 (Any non-zero number raised to the power of 0 is equal to 1)

Negative Exponent: a -n = 1/a n * (A negative exponent indicates the reciprocal of the base raised to the positive exponent)

Worked example

Example 1: Simplify: (2 3 * 2 2 ) / 2 4

Solution:

(2 3 * 2 2 ) / 2 4 = 2 (3+2) / 2 4 = 2 5 / 2 4 = 2 (5-4) = 2 1 = 2

Example 2: Simplify: (3x 2 y -1 ) 2

Solution:

(3x 2 y -1 ) 2 = 3 2 (x 2 ) 2 (y -1 ) 2 = 9x 4 y -2 = 9x 4 /y 2