Warm up Write the following expressions in exponential form b) (3x)(3x)(3x) c) 2(xy)(xy)(xy)

We are Learning to…… Use the Exponent Laws

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

Multiplying numbers in exponent form When we multiply two numbers written in exponent form and with the same base we can see an interesting result. For example: 34 × 32 = (3 × 3 × 3 × 3) × (3 × 3) = 3 × 3 × 3 × 3 × 3 × 3 = 36 = 3(4 + 2) 73 × 75 = (7 × 7 × 7) × (7 × 7 × 7 × 7 × 7) = 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 Stress that the exponents can only be added when the base is the same. = 78 = 7(3 + 5) When we multiply two numbers with the same base the exponents are added. In general, xm × xn = x(m + n) What do you notice?

Dividing numbers in exponent form When we divide two numbers written in exponent form and with the same base we can see another interesting result. For example: 4 × 4 × 4 × 4 × 4 4 × 4 = 45 ÷ 42 = 4 × 4 × 4 = 43 = 4(5 – 2) 5 × 5 × 5 × 5 × 5 × 5 5 × 5 × 5 × 5 = 56 ÷ 54 = 5 × 5 = 52 = 5(6 – 4) Stress that the exponents can only be subtracted when the base is the same. When we divide two numbers with the same base the exponents are subtracted. In general, xm ÷ xn = x(m – n) What do you notice?

Raising a power to a power Sometimes numbers can be raised to a power and the result raised to another power. For example, (43)2 = 43 × 43 = (4 × 4 × 4) × (4 × 4 × 4) = 46 = 4(3 × 2) When a number is raised to a power and then raised to another power, the powers are multiplied. In general, (xm)n = xmn What do you notice?

Using exponent laws Solve each problem by adding, subtracting and multiplying the exponents. You may choose to include negative exponents if required.

The power of 1 Find the value of the following using your calculator: 61 471 0.91 –51 01 Any number raised to the power of 1 is equal to the number itself. In general, x1 = x Because of this we don’t usually write the power when a number is raised to the power of 1.

The power of 0 Look at the following division: 64 ÷ 64 = 1 Using the second exponent law, 64 ÷ 64 = 6(4 – 4) = 60 That means that: 60 = 1 Any non-zero number raised to the power of 0 is equal to 1. Ask pupils to key any number into their calculator and raise it to the power of 0. The answer will always be 1. Ask them to raise 0 to the power of 0. An error message will be displayed. (00 can be considered as undefined. Pupils should not need to evaluate it. In fact, 00 is often defined as 0 or 1 according to the mathematical context. To prevent discontinuities, 00 = 1 would be preferred if you were plotting the graph of the form y = x0, for example, but 00 = 0 would be better for the graph y = 0x). For example, 100 = 1 3.4520 = 1 723 538 5920 = 1

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)

Negative exponents Look at the following division: 3 × 3 3 × 3 × 3 × 3 = 1 3 × 3 = 1 32 32 ÷ 34 = Using the second exponent law, 32 ÷ 34 = 3(2 – 4) = 3–2 1 32 That means that 3–2 = 1 6 1 74 1 53 Similarly, 6–1 = 7–4 = and 5–3 =

Reciprocals A number raised to the power of –1 gives us the reciprocal of that number. The reciprocal of a number is what we multiply the number by to get 1. The reciprocal of a is 1 a The reciprocal of is a b Tell pupils that if a number is written as a fraction we can easily find the reciprocal by swapping the numerator and the denominator. You could ask pupils to show why a/b × b/a will always equal 1. We can find reciprocals on a calculator using the key. x-1

Finding the reciprocals Find the reciprocals of the following: The reciprocal of 6 = 1 6 6-1 = 1 6 or 1) 6 2) 3 7 The reciprocal of = 7 3 or 3 7 = –1 Tell pupils that when we find the reciprocal of a decimal we can first write it as a decimal and then invert it. If we had a calculator we could also work out 1 ÷ the number or use the x--1 key. Improper fractions such as 7/3 can be written as mixed numbers if required. If a number is given as a decimal then its reciprocal is usually given as a decimal too. 0.8 = 4 5 The reciprocal of = 5 4 3) 0.8 = 1.25 or 0.8–1 = 1.25

Match the reciprocal pairs Establish that we have to find pairs of numbers that multiply together to make one. Each will be the reciprocal of the other. Point out that one number in the pair must be less than 1 and one number must be more than 1. Encourage pupils to convert decimals into fractions and mixed numbers into-top heavy fractions. The resulting fraction can then be reversed to find its reciprocal.

Nelson Page 222 #s 4ace, 5bdf, 6bdf, 7ace & 8bdef 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. Regular math laws for coefficients Nelson Page 222 #s 4ace, 5bdf, 6bdf, 7ace & 8bdef