Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Simplify expressions using integer and fractional exponent laws.
• Simplify expressions with both numerical and variable terms.
• Use scientific notation for calculations.
• Solve simple exponential equations.

Lesson summary

Exponents, also known as indices or powers, are a fundamental concept in mathematics. This week we're diving deeper into manipulating expressions involving exponents, building on what you've learned in earlier grades. Understanding exponents is crucial not only for success in further mathematics courses but also for modeling real-world phenomena, from population growth to compound interest. Consider, for example, the spread of information through social media – it often follows an exponential pattern. Similarly, understanding how quickly debt can accumulate with compound interest is vital for financial literacy.

Lesson notes

2.1 Revisiting the Basics An exponent indicates how many times a base number is multiplied by itself. For example, in the expression 2 3 , 2 is the base, and 3 is the exponent. This means 2 2 2 = 8. 2.2 Exponent Laws (Rules) Here's a breakdown of the exponent laws you need to know and why they work: Product Rule: a m a n = a m+n Explanation: When multiplying expressions with the same base, you add the exponents. This is because you're essentially combining the repeated multiplications. For instance, x 2 x 3 = (x x) (x x x) = x 5 .

Quotient Rule: a m / a n = a m-n (where a ≠ 0)

Explanation: When dividing expressions with the same base, you subtract the exponents. This comes from canceling out common factors in the numerator and denominator. For example, x 5 / x 2 = (x x x x x) / (x x) = x 3 .

Power of a Power Rule: (a m ) n = a mn Explanation: When raising a power to another power, you multiply the exponents. This is because you are repeating the multiplication of the base a total of n times, each time by a m . So, (x 2 ) 3 = (x 2 ) (x 2 ) (x 2 ) = (x x) (x x) (x x) = x 6 .

Power of a Product Rule: (ab) n = a n b n Explanation: When raising a product to a power, you raise each factor in the product to that power. For example, (2x) 3 = 2x 2x 2x = 2 3 x 3 = 8x 3 .

Power of a Quotient Rule: (a/b) n = a n /b n (where b ≠ 0)

Explanation: When raising a quotient to a power, you raise both the numerator and the denominator to that power. For example, (x/3) 2 = (x/3) (x/3) = x 2 /3 2 = x 2 /

9. Negative Exponent Rule: a -n = 1/a n (where a ≠ 0)

Explanation: A negative exponent indicates the reciprocal of the base raised to the positive of that exponent. This arises from the quotient rule. Consider x 2 / x 5 . Using the quotient rule, this is x -3 . But x 2 / x 5 = (xx)/(xxxxx) = 1/(xxx) = 1/x 3 .

Therefore, x -3 = 1/x 3 .

Zero Exponent Rule: a 0 = 1 (where a ≠ 0)

Explanation:* Any non-zero number raised to the power of zero is equal to

1. This also comes from the quotient rule. Think of x 2 / x 2 . This is clearly

1. Using the quotient rule, it's x 2-2 = x 0 .

Therefore, x 0 = 1. 2.3 Fractional Exponents (Rational Exponents) A fractional exponent represents a root. For example, x 1/2 is the square root of x (√x), and x 1/3 is the cube root of x ( 3 √x). In general, x m/n = n √x m = ( n √x) m .

Explanation:* Fractional exponents are another way of expressing radicals. The denominator of the fraction represents the index of the radical (the root), and the numerator represents the power to which the base is raised. 2.4 Scientific Notation Scientific notation is a way of expressing very large or very small numbers in a concise form. A number in scientific notation is written as a x 10 n , where 1 ≤ |a| m = a n , we can conclude that m = n.

Worked example

Simplify: (2x 3 y -2 ) 4

Solution:

(2x 3 y -2 ) 4 = 2 4 (x 3 ) 4 (y -2 ) 4 (Power of a Product Rule)

= 16 x 12 y -8 (Power of a Power Rule)

= 16x 12 / y 8 (Negative Exponent Rule)

Commentary: We first apply the power of a product rule, then the power of a power rule. Finally, we rewrite the expression with positive exponents using the negative exponent rule.

Simplify: (9a 4 b -2 ) 1/2

Solution:

(9a 4 b -2 ) 1/2 = 9 1/2 (a 4 ) 1/2 (b -2 ) 1/2 (Power of a Product Rule)

= √9 a 2 b -1 (Fractional Exponent and Power of a Power Rule)

= 3a 2 / b (Negative Exponent Rule)