Warm up Simplify the following expressions: a) (4x3)(3x4) b) (27x4) ÷ (3x2) c) (5x4y3)2

Use Fractional Exponents We are Learning to…… Use Fractional Exponents

Exponent notation We use exponent notation to show repeated multiplication by the same number. For example: we can use exponent notation to write 2 × 2 × 2 × 2 × 2 as Exponent or power 25 base Talk about the use of index notation as a mathematical shorthand. This number is read as ‘two to the power of five’. 25 = 2 × 2 × 2 × 2 × 2 = 32

xm × xn = x(m + n) xm ÷ xn = x(m – n) (xm)n = xmn x1 = x Exponent laws Here is a summary of the exponent laws you have met so far: xm × xn = x(m + n) xm ÷ xn = x(m – n) (xm)n = xmn x1 = x x0 = 1 (for x = 0)

Exponent laws for negative exponents Here is a summary of the exponent laws for negative exponents. x–1 = 1 x The reciprocal of x is 1 x x–n = 1 xn Stress the relationship between negative powers and reciprocals. The reciprocal of xn is 1 xn

Fractional exponents Exponents can also be fractional. Suppose we have 9 . 1 2 9 × 9 = 1 2 9 + = 1 2 91 = 9 But, 9 × 9 = 9 Because 3 × 3 = 9 In general, x = x 1 2 Similarly, 8 × 8 × 8 = 1 3 8 + + = 1 3 81 = 8 Elicit from pupils that the square root of a number multiplied by the square root of the same number always equals that number. We could also think of this as square rooting a number and then squaring it. Since squaring and square rooting are inverse operations this takes us back to the original number. The same is true of cube roots. If we cube root a number and then cube it we come back to the original number. Because 2 × 2 × 2 = 8 But, 8 × 8 × 8 = 8 3 In general, x = x 1 3

Fractional exponents What is the value of 25 ? We can think of as 25 . 3 2 We can think of as 25 . 25 3 2 1 × 3 Using the rule that (xa)b = xab we can write 1 2 25 × 3 = (25)3 = (5)3 Explain that the denominator of the power, 2, square roots the number and the numerator of the power, 3, cubes the number. = 125 In general, x = (x)m m n

Evaluate the following 1) 49 1 2 49 1 2 = √49 = 7 2) 1000 2 3 1000 2 3 = (3√1000)2 = 102 = 100 1 8 3 = 1 3√8 = 1 2 3) 8- 1 3 8- 1 3 = 1 64 2 3 = 1 (3√64)2 = 1 42 = 1 16 4) 64- 2 3 64- 2 3 = 5) 4 5 2 4 5 2 = (√4)5 = 25 = 32

Exponent laws for fractional exponents Here is a summary of the exponent laws for fractional exponents. x = x 1 2 x = x 1 n x = xm or (x)m n m

Nelson Page 229 #s 1 – 3, 5ace, 6adf & 7 Page 236 #s 4 – 6 To succeed at this lesson today you need to know and be able to use… 1. The five basic exponent laws 2. Negative exponents 3. Fractional exponents Nelson Page 229 #s 1 – 3, 5ace, 6adf & 7 Page 236 #s 4 – 6