1.what is the simplest form of the producy
sqrt 50x^7y^7 * sqrt 6 xy^4
2. What is the simplest form of the radical expression
4^3 sqrt 3x + 5^3 sqrt 10x
3. What is the simplest form of the radical expression
sqrt 2 + sqrt5 / sqrt 2 - sqrt 5
if someone could explain the steps it would be helpful cause I have missed these on my homework and would like to know how to do them
Can someone please help me out.
what is the simplest form of the radical expression 4 3 square root 3 x t 5 3 square root 10 x
1. To simplify the expression sqrt(50x^7y^7) * sqrt(6xy^4), we can combine the two radicals by multiplying the numbers under the square root and the variables with the same exponents.
First, simplify each expression under the square root:
- sqrt(50x^7y^7) = sqrt(25 * 2 * x^6 * x * y^6 * y)
= 5 * x^3 * y^3 * sqrt(2xy)
- sqrt(6xy^4) = sqrt(2 * 3 * x * y^4)
= y^2 * sqrt(2xy)
Now, multiply the two simplified expressions together:
(5 * x^3 * y^3 * sqrt(2xy)) * (y^2 * sqrt(2xy))
Multiplying the coefficients (5 * 1) gives 5. Multiplying the variables with the same exponents gives x^3 * x = x^4 and y^3 * y^2 = y^5.
(5 * x^3 * y^3 * sqrt(2xy)) * (y^2 * sqrt(2xy)) = 5x^4y^5 * sqrt(2xy * 2xy)
Finally, simplify the expression under the square root:
sqrt(2xy * 2xy) = sqrt(4 * x^2 * y^2)
= 2xy
Therefore, the simplest form of the product is 5x^4y^5 * 2xy = 10x^5y^6.
2. To simplify the expression 4^3 sqrt(3x) + 5^3 sqrt(10x), we can simplify each radical expression separately, and then combine them using basic addition.
First, simplify each expression under the square root:
- 4^3 sqrt(3x) = 4 * 4 * 4 * sqrt(3x) = 64sqrt(3x)
- 5^3 sqrt(10x) = 5 * 5 * 5 * sqrt(10x) = 125sqrt(10x)
Now, we can combine the two radical expressions:
64sqrt(3x) + 125sqrt(10x)
Since the two terms have different radicals, we cannot simplify any further. Therefore, the simplest form of the expression is 64sqrt(3x) + 125sqrt(10x).
3. To simplify the expression (sqrt(2) + sqrt(5)) / (sqrt(2) - sqrt(5)), we can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.
The conjugate of (sqrt(2) - sqrt(5)) is (sqrt(2) + sqrt(5)). Therefore, we multiply the numerator and denominator by (sqrt(2) + sqrt(5)):
(sqrt(2) + sqrt(5)) / (sqrt(2) - sqrt(5)) * (sqrt(2) + sqrt(5)) / (sqrt(2) + sqrt(5))
Applying the FOIL method to the numerator and denominator, we get:
[(sqrt(2))^2 + sqrt(2)sqrt(5) + sqrt(2)sqrt(5) + (sqrt(5))^2] / [(sqrt(2))^2 - (sqrt(5))^2]
Simplifying, we have:
(2 + 2sqrt(10) + 5) / (2 - 5)
This becomes:
(7 + 2sqrt(10)) / (-3)
Finally, we can simplify further by dividing both the numerator and denominator by -1:
- (7 + 2sqrt(10)) / 3
Therefore, the simplest form of the expression is -(7 + 2sqrt(10)) / 3.