hi I am teacher Daisy now let's learn form three chapter one indices first we learn index notation or index form let's say there is a a ^ then so a is the base and n is the index or exponent if five ^ 2 5 is the basin 2 is the index or exponent number one right repeated multiplication and index form the value of index is the same as the number of times bases multiplied repeatedly where a is not equal to zero example a since the base 5 repeatedly multiplied two times so the base will be 5 and index will be 2 example B since the base 5 repeatedly multiplied 3 times so the base will be 5 and index will be 3 example C since the base 0.4 repeatedly multiplied 2 times so the base will be 0.4 an index will be 2 example D since the base 1/4 repeatedly multiplied 4 times so the base will be in parentheses 1/4 an index will be 4 number two convert numbers or algebraic terms and index form into repeated multiplications example a in parentheses negative P to the power of 7 equals in parentheses negative P repeated multiplied seven times example B in parenthesis one end to the power of three equals in parentheses one and repeated multiplied three times number three in order to convert a number into a number in index form there are two methods repeated division and repeated multiplication example a right 16 an index form using base of two using repeated division method 16 divided by two equals eight eight divided by two equals four division continued until one is obtained the number of divisions is four hence 16 equals two to the power of four using repeated multiplication method two times two equals four four times two equals eight eight times two equals 16 hence 16 equals two to the power of for example B right 32 3125 an index form using base of two-fifths using repeated division method divide the numerator 32 divided by two equals 16 16 divided by 2 equals eight division continued until one is obtained the number of divisions is five divide the denominator 3000 125 divided by five equals 625 625 divided by five equals 125 division continued until one is obtained the number of divisions is five hence 32 3125 equals in parentheses two-fifths to the power of five using repeated multiplication method 2/5 times two fifths equals 425 425 times 2/5 see 125 multiplication continued until 32 3125 is obtained the number of multiplication is 5 hence 32 3125 equals in parentheses 2/5 ^ 5 number 4 determine the value of number in index form a to the power of n example a 2 to the power of 5 using multiplication method 2 times 2 equals 4 4 times 2 equals 8 8 times 2 equals 16 16 times 2 equals 32 hence 2 to the power of 5 equals 32 using scientific calculator press 2 to the power of 5 then press equals will get 32 example be the repeated multiplication method can use to solve this problem just same as the previous example using scientific calculator press Open bracket negative four close bracket to the power of 3 then press equals we'll get negative 60 for example C similarly the repeated multiplication method can use to solve this problem using scientific calculator press open bracket four-fifths closed bracket to the power of 4 then press equals we'll get 256 625 reminder negative or fractional base must be placed within brackets when using calculator to calculate values of given numbers let's look at an example of the difference of within without bracket in parentheses negative 2 to the power of 4 will get 16 while negative 2 to the power of 4 we'll get negative 16 now we learn the five law of indices first multiplication of number and index form with same base a to the power of M times a to the power of n equals a to the power of M plus n for your information when we write a is implicitly means de to the power of besides that negative eight of the power is not equal to in parentheses negative a to the power of n as discussed earlier example a two square times two cubed equals two to the power of two plus three which will get two to the power of five by using repeated multiplication can prove the law example B group the coefficients and the algebraic terms with the same base then do the operations of the coefficients three times one sixth times twelve and we'll get six after that apply the first law of indices for the algebraic number M to the power of four plus five plus one lastly the answer will be six and M to the power of ten example C group the terms with the same base then do addition operation of the indices for the terms with the same base at last we'll end up M to the power of seven and n to the power of seven second law division of numbers in index form with the same base a to the power of M divided by a to the power of n equals a to the power of M minus n example a by applying second law of indices to to the power of six minus two equals two to the power of four using repeated multiplication two divided by two and again two divided by two will get the same result example B group the numbers and algebraic terms with the same base first operate the coefficient 25 fifths and apply second law of indices on X&Y at last we'll end up as 5x y square third law raise a number in index form to a power example a three squared raised to the power of four the index two times four equals eight we'll end up three to the power of eight using repeated multiplication can prove that 3 squared repeatedly multiplies four times will become three to the power 2 plus 2 plus 2 plus two example be H cubed raised to the power of 10 the index 3 times 10 equals 30 we'll end up H to the power of 30 use law of indices to perform operations of multiplication and division first the power Q times with all indices M and n will end up a to the power of M Q and B ^ NQ the power Q times with all indices M and n will end up a to the power of them Q divided B to the power of n Q example a the power 4 times all the indices inside the bracket and we'll get the answer example B the power of three times all the indices of numbers and algebraic terms inside the bracket the term do not write any index is actually index one after that do the division operation the division of algebraic terms with same base will be minus of indices fourth law a ^ 0 equals 1 a to the power of negative n a equals 1 over a to the power of n where a is not 0 example a two to the power of zero equals one and using repeated multiplication can prove that example B two to the power of negative 2 equals 1 over 2 squared by using repeated multiplication can prove that example see state the term in positive index form X to the power of negative 4 equals 1 over X to the power of 4 - 8 to the power of negative N equals 2 over a to the power of n there in mind that it is not equals 1 over 2 a to the power of n example D state the term and negative index form 1 over 3 to the power of 4 equals 3 to the power of negative 4 in parentheses 4/5 to the power of 8 equals in parentheses 5/4 to the power of negative 8/5 law relationship between fractional indices and roots and powers and root of a equals a to the power of 1 over n where a is not 0 1 over a is the reciprocal of a example--a in order to get the value of x there are two methods method one using square roots square roots are used to eliminate squares in an equation if we square roots left-hand side we need two square roots right-hand side as well nine is three squared square roots of x squared equals square roots of three squared so we'll get x equals three method 2 using reciprocal the reciprocal of 2 is 1/2 multiplied the indices of left hand side and right hand side with 1/2 we'll get x equals 3 example b method 1 using cube roots cube roots are used to eliminate cube in an equation if we cube roots left hand side we need to cube roots right hand side as well 64 is 4 cubed cube roots of x cubed equals cube roots of 4 cubed so we'll get x equals 4 method 2 using reciprocal the reciprocal of 3 is 1/3 multiplied the indices of left hand side and right hand side with 1/3 we'll get x equals 4 example see convert the term into form a to the power of 1 over n cube roots of negative 27 equals in parentheses negative 27 to the power of 1/3 fifth root of M equals M to the power of 1/5 example D convert the term into the form into the form n root of a in parentheses negative 1,000 to the power of 1/3 equals cube roots of negative 1,000 and to the power of 1/12 equals 12th roots of n the formula in the rounded rectangle shows that a to the power of M over and can be written in different forms example a convert the following in to specify forms by comparing with the formula we know that 27 is a 2 is M and 3 is n in this example example B convert the following into the specify forms by comparing with the formula we know that M s a2 is M and v is n in this example example C calculate value of term fifth root of negative 32 first write it in parentheses negative 32 to the power of 1/5 in order to simplify this change negative 32 to negative 2 to the power of 5 multiplication of the indices will get 1 finally the answer will be negative 2 exercises thanks for watching if you find this video helpful don't forget to Like share and subscribe our channel and if you got any question can comment below