Hi, could someone please tell me how I would go about working these out: (the answer must be in the form a(sqroot)b where a and b are integers and b is as small as possible.)

1) Simplify (sqroot)500
2) Simplify (sqroot)50 x (sqroot)60

Also..

Simplify the following fractions so that there is no root in the denominator:
1/(sqroot)5
12/(sqroot)3

Thankyou so much!

1) sqrt500 =sqrt(25 x4 x 5)
=(sqrt 25)x(sqrt4)x (Sqrt5)
=5 x 2 x (sqrt5)
=10sqrt5
therefore a=10 and b=5

2) sqrt50 xsqrt60 =Sqrt300
=sqrt25 x sqrt4 x sqrt3
=5 x 2 x sqrt3
=10sqrt3
therefore a=10 and b=3

To simplify the expression (sqrt)500, we can start by factoring 500 into its prime factors: 500 = 2^2 x 5^3.

Since the square root of both 2^2 and 5^2 are integers (2 and 5, respectively), we can take them out of the square root:
(sqrt)500 = (sqrt)(2^2 x 5^2 x 5)

Using the properties of square roots, we can split the square root into two separate square roots:
(sqrt)500 = (sqrt)(2^2) x (sqrt)(5^2) x (sqrt)5

Simplifying, we get:
(sqrt)500 = 2 x 5 x (sqrt)5 = 10(sqrt)5

Therefore, the simplified form of (sqrt)500 is 10(sqrt)5, where a = 10 and b = 5.

Similarly, to simplify the expression (sqrt)50 x (sqrt)60, we can factor both numbers:
50 = 2^2 x 5
60 = 2^2 x 3 x 5

Again, we can separate them into two square roots:
(sqrt)50 x (sqrt)60 = (sqrt)(2^2 x 5) x (sqrt)(2^2 x 3 x 5)

Simplifying, we get:
(sqrt)50 x (sqrt)60 = 2 x 5 (sqrt)2 x (sqrt)3 x (sqrt)5 = 10(sqrt)2(sqrt)3(sqrt)5

Since we want to minimize the value of b, we should choose the smallest possible value of b. In this case, the smallest value of b is 3.

Therefore, the simplified form of (sqrt)50 x (sqrt)60 is 10(sqrt)3, where a = 10 and b = 3.

Lastly, to simplify the fractions 1/(sqrt)5 and 12/(sqrt)3 where there is no square root in the denominator, we can use a technique called rationalizing the denominator.

For 1/(sqrt)5, we can multiply both the numerator and denominator by (sqrt)5:
1/(sqrt)5 x (sqrt)5/(sqrt)5 = (sqrt)5/5

Therefore, the simplified form of 1/(sqrt)5 is (sqrt)5/5.

For 12/(sqrt)3, we can multiply both the numerator and denominator by (sqrt)3:
12/(sqrt)3 x (sqrt)3/(sqrt)3 = 12(sqrt)3/3

Therefore, the simplified form of 12/(sqrt)3 is 12(sqrt)3/3.

I hope this explanation helps! Let me know if you have any further questions.