23112020 Index laws MATHSWATCH CLIP 34 35 GRADE

23/11/2020 Index laws MATHSWATCH CLIP 34, 35 GRADE 2 L. O. To be able to § Simplify expressions involving multiplication and division of terms § Simplify expressions involving brackets raised to a power Key Words index, power, multiply, divide, product, brackets, Starter

23/11/2020 Index laws MATHSWATCH CLIP 34, 35 GRADE 2 CONTENTS – Click to go to… MULTIPLYING DIVIDING BRACKETS FRACTIONAL AND NEGATIVE

23/11/2020 Index laws MATHSWATCH CLIP 34, 35 GRADE 2 MULTIPLYING

23/11/2020 MATHSWATCH CLIP 34, 35 GRADE 2 Index laws Simplify: x + x + x= 5 x x to the power of 5 Simplify: x × x × x = x 5 as been written using index notation. xn The number x is called the base. The number n is called the index or power.

23/11/2020 Index laws MATHSWATCH CLIP 34, 35 GRADE 2 We can use index notation to simplify expressions. For example, 3 p × 2 p = 3 × p × 2 × p = 6 p 2 q 2 × q 3 = q × q × q = q 5 3 r × r 2 = 3 × r × r = 3 r 3

23/11/2020 Index laws MATHSWATCH CLIP 34, 35 GRADE 2 When we multiply two terms with the same base the indices are added. For example, a 4 × a 2 = (a × a × a) × (a × a) = a × a × a × a = a 6 = a (4 + 2) In general, xm × xn = x(m + n)

Index laws 23/11/2020 2 5 a ×a = 2 3 a × 4 a 5 MATHSWATCH CLIP 34, 35 GRADE 2 a×a×a×a =a 2+5 =a 7 = What about but Add the powers these ? the MULTIPLY ‘numbers’. 3×a×a× 4×a×a×a = 12 a 2+5 = 12 a 7 m n a ×a = a m+n

Index laws 23/11/2020 3 MATHSWATCH CLIP 34, 35 GRADE 2 2 4 a × 2 a b =4×a×a×a× 2×a×a×b 3+2 = 8 a b 5 = 8 a b 3 2 5 a b × 3 ab c =5×a×a×a×b× 3×a×b×b×c = 15 a 3+1 1+2 b c 4 3 = 15 a b c m n a ×a = a m+n

23/11/2020 Index laws MATHSWATCH CLIP 34, 35 GRADE 2 MULTIPLYING terms § Treat any “normal numbers” as normal – MULTIPLY THEM § Treat any “powers” using index laws – ADD THEM § Remember – if it looks like there is no power there, it’s an invisible 1

23/11/2020 Index laws MATHSWATCH CLIP 34, 35 GRADE 2 Examples (a) 6 × 3 x (b) 5 y × 4 (c) 3 x × 2 x (d) x × x 2 (e) 2 a × 3 b (f) xy × 3 y = 18 x = 20 y = 6 x 2 = x 3 = 6 ab = 3 y 2 x (g) 3 a × 3 a 2 (h) 5 p × 2 p × p (i) 4 y 2 × 3 y (j) -2 × 3 x 2 (k) -5 a × 3 a (l) -4 x 2 y × xy 2 = 9 a 3 = 10 p 3 = 12 y 3 = -6 x 2 = -15 a 2 = - 4 x 3 y 3

23/11/2020 MATHSWATCH CLIP 34, 35 GRADE 2 Index laws Simplify a 2 6 h 5 12 w 2 6 a 2 x 3 5 a 3 b 4 a 2 b 2 c 2 a 3 bcd 6 a 3 b m 8 t 2 36 a 4 b 2 c 1

23/11/2020 Index laws MATHSWATCH CLIP 34, 35 GRADE 2 Task – Simplify each expression as much as possible (a) 6 x 8 p (b) 5 m x 3 (c) 2 x x 7 x (d) x 2 x (e) 4 p x 3 q (f) 3 xy x 2 x = 48 p = 15 m = 14 x 2 = 2 x 3 = 12 pq = 6 x 2 y (g) m x 3 m 2 (h) 3 a x 2 b 2 x c (i) y 2 x 7 y (j) 4 x -3 x 2 (k) -3 a 2 x 3 a (l) -2 p 2 x pq 2 = 3 m 3 = 6 ab 2 c = 7 y 3 = -12 x 2 = -9 a 3 = - 2 p 3 q 2

23/11/2020 Index laws Try the questions on the worksheet MATHSWATCH CLIP 34, 35 GRADE 2

23/11/2020 Index laws MATHSWATCH CLIP 34, 35 GRADE 2

23/11/2020 Index laws DIVIDING MATHSWATCH CLIP 34, 35 GRADE 2

Index laws 23/11/2020 MATHSWATCH CLIP 34, 35 GRADE 2 Remember, in algebra we do not usually use the division sign, ÷. Instead, we write the number or term we are dividing by underneath like a fraction. For example, (a + b) ÷ c is written as a + b c

MATHSWATCH CLIP 34, 35 GRADE 2 Index laws 23/11/2020 As with fractions, we can often simplify expressions by cancelling. For example, 2 6 p 6 p 2 ÷ 3 p = 3 p 3 n n 3 ÷ n 2 = 2 n 1 1 2 1 n × n = n × n 6 × p = 3 × p = n = 2 p 1 1

Index laws 23/11/2020 MATHSWATCH CLIP 34, 35 GRADE 2 When we divide two terms with the same base the indices are subtracted. For example, a 5 ÷ a 2 = a × a × a = a 3 = a (5 – 2) a × a 2 4 p 6 ÷ 2 p 4 = 4 × p × p × p = 2 p 2= 2 p(6 – 4) 2 × p × p In general, xm ÷ xn = x(m – n)

MATHSWATCH CLIP 34, 35 GRADE 2 Index laws 23/11/2020 Task - Simplify 5 3 3 3 4 1 1 1 3 2 2 2 1 1 2 5 2 4 3 1 4 1 4 6 3 1

MATHSWATCH CLIP 34, 35 GRADE 2 Index laws 23/11/2020 Examples a 6 ÷ a 2 = a 4 z ÷z =z 1 must 1 subtract 0 the Clearly you But this is pointless ! powers but DIVIDE the numbers ! 8 a 6 ÷ 2 a 2 = 4 a 4 10 w 5 y 3 ÷ 5 w 2 y 1 = 16 a 3 b 2 2 ab = m = 1 2 w 3 y 2 8 a 2 b n 20 h 3 y 2 z = 4 hyz m-n a ÷ a = a 5 h 2 y × 1

11/23/2020 Index laws MATHSWATCH CLIP 34, 35 GRADE 2 DIVIDING terms § Treat any “normal numbers” as normal – DIVIDE THEM § Treat any “powers” using index laws – SUBTRACT THEM § Remember – if it looks like there is no power there, it’s an invisible 1

23/11/2020 Index laws Task - Simplify MATHSWATCH CLIP 34, 35 GRADE 2 1) a 7 ÷ a 4 2) 9 a 4 ÷ 3 a 3 a 3 3) 12 w 3 y 2 ÷ 6 wy 2 w 2 y 4) 20 h 3 x ÷ 4 h 2 x 5 h 5) 16 a 4 b 3 c ÷ a 2 b 316 a 2 c 6) 25 x 2 y ÷ 5 x 35 x y -1 or 5/x 7) a 3 b 2 ab 14 wh 3 7 wh 2 8) 9) a 3 a 2 b 2 h 12 x 3 y 4 x 2 10) 18 a 3 b 2 c 3 9 ab 2 c 4 3 xy 2 a 2 c-1 or 2 a 2/c

23/11/2020 Index laws MATHSWATCH CLIP 34, 35 GRADE 2

23/11/2020 Index laws MATHSWATCH CLIP 34, 35 GRADE 2

23/11/2020 Index laws BRACKETS MATHSWATCH CLIP 34, 35 GRADE 2

23/11/2020 Index laws MATHSWATCH CLIP 34, 35 GRADE 2 Sometimes terms can be raised to a power and the result raised to another power. For example, (y 3)2 = y 3 × y 3 (pq 2)4 = pq 2 × pq 2 = (y × y) × (y × y) = p 4 × q (2 + 2 + 2) = y 6 = p 4 × q 8 = p 4 q 8

23/11/2020 Index laws MATHSWATCH CLIP 34, 35 GRADE 2 When a term is raised to a power and the result raised to another power, the powers are multiplied. For example, (a 5)3 = a 5 × a 5 = a(5 + 5) = a 15 = a(3 × 5) In general, (xm)n = xmn

MATHSWATCH CLIP 34, 35 GRADE 2 Index laws 23/11/2020 ( a 2 )3 = a 6 So you square, cube etc. . the ‘numbers’ but you MULTIPLY THE POWERS ! ( a × a )3 a × a × a × a = a 6 ( 5 a 4 )2 = 25 a 8 (5×a×a )2 m n (a ) = a m n mn ( ba ) = b a 5×a×a× 5×a×a = 25 a 8

Index laws 23/11/2020 Examples ( a 4 )3 = a 12 ( 4 w 3 )2 = 16 w 6 ( 1 3 2 5 a b ) = 125 a 6 b 3 ( 3 x 3 y 4 )2 = 9 x 6 y 8 m n (a ) = a m n mn ( ba ) = b a MATHSWATCH CLIP 34, 35 GRADE 2

23/11/2020 Index laws Basic index laws - SUMMARY MATHSWATCH CLIP 34, 35 GRADE 2 Here is a summary of the index laws you have met so far: xm × xn = x(m + n) xm ÷ xn = x(m – n) (xm)n = xmn x 1 = x x 0 = 1 (for x ≠ 0)

Index laws 23/11/2020 MATHSWATCH CLIP 34, 35 GRADE 2 Task – Re-write the following without brackets 1) (2 a 2)3 = 8 a 6 2) (m 3 n)4 = m 12 n 4 3) (t– 4)2 = t– 8 4) (3 g 5)3 = 27 g 15 5) (ab– 2)– 2 = a– 2 b 4 6) (p 2 q– 5)– 1 = p– 2 q 5 7) (h½)2 = 8) (7 a 4 b– 3)0 = h 1

23/11/2020 MATHSWATCH CLIP 34, 35 GRADE 2 Index laws Task – Re-write these expressions without any brackets 1) ( a 2 )5 a 10 2) ( 3 x )2 3) ( 5 y 2 )3 125 y 6 4) ( 2 a 2 b )4 16 a 8 b 4 5) ( 3 x 2 y 3 )2 9 x 4 y 6 6) ( 4 a 0 b)2 16 b 2 7) ( a-2 )3 8) ( x-1 y-2 )-3 x 3 y 6 a-6 or 1/a 6 9 x 2

23/11/2020 Exam style question Index laws MATHSWATCH CLIP 34, 35 GRADE 2

Index laws 23/11/2020 MATHSWATCH CLIP 34, 35 GRADE 2 Task - Simplify 1 y 2 x y 3 y 5 2 2 y 2 x 3 y 4 6 y 6 3 5 p 2 x 3 p 3 x 2 p 30 p 6 4 8 k 3 x 2 k-4 x 3 k 2 48 k 5 ab 2 x a 2 b 3 x a 2 b 4 a 5 b 9 6 2 a 3 b 2 x 3 ab 4 x 2 a 2 b 2 12 a 6 b 8 7 (2 pq 2)2 = 2 pq 2 x 2 pq 2 = 4 p 2 q 4 8 (3 a 2 b 3)2 = 3 a 2 b 3 x 3 a 2 b 3 = 9 a 4 b 6 9 (5 m 2 n 3)2 = 5 m 2 n 3 x 5 m 2 n 3 = 25 m 4 n 6 10 (2 pq 2)3 = 2 pq 2 x 2 pq 2 = 8 p 3 q 6 11 (3 a 2 b 3)4 = 81 a 8 b 12 (2 m 2 n 3)5 = 32 m 10 n 15 Raise the number to the given power and multiply the indices.

MATHSWATCH CLIP 34, 35 23/11/2020 GRADE 2 Task – Colour by numbers. Simplify each section and choose the correct colour. Index laws

23/11/2020 Index laws MATHSWATCH CLIP 34, 35 GRADE 2 FRACTIONAL AND NEGATIVE INDICES

Index laws 11/23/2020 MATHSWATCH CLIP 34, 35 GRADE 2 Complex index laws - SUMMARY Here is a summary of the index laws for fractional and negative indices: 1 x 1 –n x = xn The reciprocal of x is 1 x x– 1 = The reciprocal of xn x = x 1 2 n x = x 1 n m n n n x = xm or ( x)m 1 is n x

23/11/2020 Index laws Examples -1 2 a means MATHSWATCH CLIP 34, 35 GRADE 2 2 a -1 -1 1 ( 2 a ) means ( 2 a ) -1 -2 3 a means -2 ( 3 a ) means -2 3 a ( 3 a 1 ) -2 or 1 2 9 a

MATHSWATCH CLIP 34, 35 GRADE 2 Index laws 23/11/2020 Task – Evaluate the following 1) 4) 2 -3 5 -2 1/8 1/25 2) 3 5) -1 5 1/3 0 1 3) 6) 3 -3 10 1/27 -4 1/10000 If a = 2, b = 3, c = 4 evaluate the following : ( Leave your answers as fractions in their simplest form ) -4 7) a 10) ab 13) ( abc ) 16) a -2 -b -1 -2 1/16 8) b 2/9 11) ( 2 ab ) 1/24 14) 6 ( bc ) 17) -a 1/8 cb -2 -1 0 9) c 1/144 12) abc 1/2 15) a ( bc ) 18) -2 a 1/9 4/9 1 -1 ( 2 a ) 1 1/2 -2 1/72 1/256

MATHSWATCH CLIP 34, 35 GRADE 2 Index laws 23/11/2020 Example c a 3 8 b 2 = a = 8 = 4 b c 2 3 power root Combinations The cube of rootsofand 8 powers is 2 and can be written then 2 using a fractional squared index. is 4.

MATHSWATCH CLIP 34, 35 GRADE 2 Index laws 23/11/2020 Task – Evaluate the following if a = 2, b = 3 and c = 4 1) 4) 9 ba (ab)a 7) 36 10) ca-1 -1 a 2) 36 5) 1/6 8) 2 11) c-a c b a 2 c 1/16 3) 8 6) ab × ac b -b 128 1/27 b c b 81 9) ( bc )-1 1/12 9 b-a 1 b 2 a - c 1 12)