Question

What is `sqrt121 + root3 343`?

Answer 1

`sqrt(121)+root(3)(343) = 18`

Explanation:

`sqrt(121) = 11 hArr 11^2=121`
`root(3)(343)= 7 hArr 7^2=343`

`sqrt(121)+root(3)(343) = 11 + 7 = 18`

Answer 2

18

Explanation:

Remember that to get out of roots without a calculator you must factor the numbers within the roots with prime numbers. Once you have the same number of a particular prime as the "root" number you can take that number out of the root until you have nothing inside OR you leave the odd ones in

For example `sqrt(9)` 9 is 3 times 3, so 2 threes, therefore we can take 1 three out and are left with nothing so the answer would be 3. Now if we took `sqrt(18)`, 18 is `2*3*3` so we could take the 3 out, but the two is left in, so it is equal to `3sqrt(2)` as another example, take `root(3)(250)`, 250 is `2*5*5*5` so we can take the 5s out but leave the 2 in for a simplified `5root(3)(2)`

THIS case is easier as `sqrt(121) = 11 and root(3)(343) = 7`
`i.e. 121 = 11*11 and 343 = 7*7*7`
so `11+7=18`

Answer 3

18`" "`Trial and error demonstrated using approximations.

Explanation:

Given expression: `" sqrt(121)+root(3)(343)`

`color(blue)("Not using a calculator")`

`color(brown)("To find the value of "sqrt(121))`

Known: `"From multiplication tables "11xx11=121 -> sqrt(121)=11`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(brown)("To find the value of "root(3)(343))`

`color(purple)("Lets look at 343 using trial and error initially, but with a bit of forethought.")`

1st point:
we need to end up in the hundreds so we need numbers that are going to end up where the first multiplication will a value in the tens

`color(green)("Step 1")`

Lets look at one we know: `5xx5=25" " and " "5xx25=125`. So we need to be higher than 5.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(green)("Step 2")`
`6xx6=36" but "3xx6=18" but the 3 is in tens so "10xx18=180`

This is not what we are looking for as the final value is likely to be close to 200
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(green)("Step 3")`
Lets look at 7

`7xx7=49" which is nearly 50 and "10xx5xx7=350`. As this estimate is close to 343 the value of 7 could be the one we want. lets try it.

`7xx7=49`

`color(purple)("Notice the way I 'split' the numbers to make multiplication easier to do in your head.")`

`49xx7= (9xx7)+(4xx7xx10)=63+280= 343" "`

`color(purple)("'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")`
`color(green)("Answer")`

`color(green)(" "sqrt(121)+root(3)(343)" "=" " 11+7" "=" "18)`