for example number three we're going to approximate the square root of 11 to the one decimal place our first step for this question is to find two integers that the square root of 11 is in between so we have a squared of B squared and again it helps to know our perfect squares so we have 2 squared which is equal to 4 3 squared is nine and then 4 squared is 16. so that means that the square root of 11 has to be between 3 squared and 4 squared so let's write that out so we have the square root of 3 squared which is 9. and then the square root of 11 and then square root of 16. so now as we look between what units are in between 9 11 and 16 so there are 2 between 9 and 11 and 5 in between 11 and 16 so that means that we can use a ratio to determine what our decimal is going to be so we take our first number which is 2. and then we can use a ratio to find how far it's going to go in between 9 and 16 so then we can divide by two plus five so we get 2 divided by 7 which if we round to one decimal place will be 0.3 so now if we go back to this we can rewrite this as 3 and then root 11 and then four so our first digit is going to be three since as it's not going to get to 4 yet but it's greater than 3 and then our decimal will be 0.3 that we found using a ratio and so we can approximate the square root of 11 as 3.3 now let's do the same for approximating the cube root of 140 to 1 decimal place so we'll start off by writing that the cube root of 140 is between a cubed and B Cubed and then we can use our knowledge of perfect cubes to figure out which two numbers are the cube root of 140 is in between so we have 2 cubed is eight three cubed is 27 4 cubed is 64. 5 cubed is 125 and 6 cubed is 216. so now that we know that 140 is going to fall in between 125 and 216. we can rewrite this as the cube root of 125 is less than the cube root 40 Which is less than the cube root of 2 16. so now we can figure out how many units are in between these digits and we have 15 and 76. so then again we take our first number which is 15. and we divide it by the sum of the two which is 15 and 76. so we get 15 over 91 which is approximately 0.2 then we can go back to this and rewrite this as 5. and then we have cube root of 140 is less than six and so our first digit will be five since that's the lower limit and then the decimal will be 0.2 and so we can approximate the cube root of 140 as 5.2