sqrt (x^5 y^2) +sqrt(9xy^2)
To simplify the expression sqrt(x^5 y^2) +sqrt(9xy^2), we can apply the properties of square roots.
First, let's simplify each term separately:
1. sqrt(x^5 y^2):
To simplify this term, we need to break down the expression inside the square root into its prime factors.
- The prime factorization of x^5 is x * x * x * x * x.
- The prime factorization of y^2 is y * y.
Now we can rewrite sqrt(x^5 y^2) as sqrt((x * x * x * x * x) * (y * y)).
According to the property of square roots, sqrt(a * b) = sqrt(a) * sqrt(b), where a and b are positive numbers.
Using this property, we can split the square root expression as sqrt(x * x * x * x * x) * sqrt(y * y).
The simplified version becomes x^2 * y.
2. sqrt(9xy^2):
We can similarly simplify this term.
- The prime factorization of 9 is 3 * 3.
- The prime factorization of xy^2 is x * y * y.
Now we can rewrite sqrt(9xy^2) as sqrt((3 * 3) * (x * y * y)).
Using the property of square roots, we can split the square root expression as sqrt(3 * 3) * sqrt(x * y * y).
The simplified version becomes 3 * y.
Now, we can substitute these simplified terms back into the original expression:
sqrt(x^5 y^2) + sqrt(9xy^2) = x^2 * y + 3 * y = x^2 * y + 3y.
Therefore, the simplified form of the expression sqrt(x^5 y^2) + sqrt(9xy^2) is x^2 * y + 3y.