Are there any perfect squares left under the radical 33?

Question

Write no perfect squares under the radical.
1. radical of 300
2. radical of 3/121

Answer 1

sqrt(300) = sqrt(100 * 3) = sqrt(10*10 * 3) = 10*sqrt(3).
sqrt(3/121) = sqrt(3/(11*11)) = sqrt(3)/11.

No, 33 is squarefree - it is 3*11, which contains no squares.

Answer 2

there could be radical of ( 1 x 33) so, a perfect square is 1 times radical ( 33)

Answer 3

Actually, there is a way to "force" any perfect square we want under the radical sign. Sometimes this little trick comes in handy if we are wanting a fairly close approximation to the square root of a small number.

The idea here is to get a quotient left under the radical sign which is fairly close to 1. Then we can use our knowledge of mathematics and reason to find a close approximation to the square root of the original number. For example, when we take the square root of 33, we can divide it by the closest perfect square which is just smaller than it. So, 33 = 25 x 33/25.

When we take its square root, we can extract the square root of 25, and we are left with 33/25 under the radical sign. When we do this, we get:

sq. rt. 33 = 5 x sq. rt. (33/25).

Now let's divide out this last fraction. We get 1.32. We recall that 1.10 squared is 1.21 and 1.20 squared is 1.44. Now 1.32 is just slightly less than halfway between 1.21 and 1.44. We surmise that the square root of 1.32 therefore is probably slightly less than halfway between the square roots of these two numbers. Indeed, we could interpolate between the values by setting up this ratio:

0.11/0.23 = x/0.10,

where 0.11 is the difference between 1.21 and 1.32, 0.23 is the difference between 1.21 and 1.44, x is the difference between the square root of 1.21 and 1.32 and 0.10 the difference between the square root of 1.21 and 1.44.

We calculate x: x = (0.11/0.23) x 0.10 = 0.0478.

We add this to the square root of 1.21, to get 1.1478, which we round to two decimal places, since the other numbers are accurate to that number of digits, to get 1.15.

Now we multiply this by 5 to get 5.75, an approximation to the square root of 33. Squaring 5.75, we get 33.06, which is pretty accurate for guesswork and old fashioned reasoning and calculation.

Answer 4

1. 10sqrt(3)
2. [sqrt(3)]/11