Given x > 0 express in simplest radical form with a rational denominator.
√336x^12/√7x^4
A hyperbola is defined by the equation (y-4)^2/25 - (x-8)^2 = 1
what is the co-vertices,
The standard equation of a hyperbola with horizontal transverse axis is:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
where (h, k) is the center of the hyperbola, a is the distance from the center to each vertex along the x-axis, and b is the distance from the center to each vertex along the y-axis.
Comparing the given equation to the standard equation, we can see that this hyperbola has a vertical transverse axis (because the x term is negative). So we have:
(h, k) = (8, 4)
a^2 = 1 (from the coefficient of x^2 being -1)
b^2 = 25 (from the coefficient of y^2 being 1)
So the distance from the center to each vertex along the y-axis is b = 5. The co-vertices are the points that are distance b from the center and lie along the minor axis (i.e. along the x-axis). So we need to find the x-coordinates of these points. They are given by:
x = h ± a = 8 ± 1
Therefore, the co-vertices are the points (7, 4) and (9, 4).
To simplify the expression √336x^12/√7x^4 with a rational denominator, we can separate the square roots and simplify each one individually.
First, let's simplify the numerator. We have √336x^12. To simplify the square root, we can break down 336 into its prime factors:
336 = 2 * 2 * 2 * 2 * 3 * 7
Inside the square root, we can pair up the prime factors in groups of two:
√336 = √(2 * 2 * 2 * 2 * 3 * 7)
= √(2^2 * 2^2 * 3 * 7)
= √(2^4 * 3 * 7)
Taking the square root of each group, we can bring out the perfect squares from under the radical sign:
√(2^4 * 3 * 7)
= 2^2 * √(3 * 7)
= 4√(3 * 7)
= 4√21
Now let's simplify the denominator. We have √7x^4. Since x > 0, we can omit the absolute value signs for the square root of x^4:
√7x^4 = √(7 * x^2 * x^2)
= √(7 * x^2)^2
= 7x^2
Putting it all together, we have:
√336x^12/√7x^4 = (4√21 * x^6)/(7x^2)
To express it in the simplest radical form, with a rational denominator, we can cancel out common factors between the numerator and denominator:
(4√21 * x^6)/(7x^2)
= (4/7) * (√21 * x^6/x^2)
= (4/7) * (√21 * x^(6-2))
= (4/7) * √(21x^4)
Therefore, the simplified expression with a rational denominator is:
(4/7) * √(21x^4)
Simplify the expression under the radical in the numerator:
√[336x^12] = √[(16x^6)(21x^6)] = √[16x^6] √[21x^6] = 4x^3√21
Simplify the expression under the radical in the denominator:
√[7x^4] = √[7] √[x^4] = √[7] x^2
Now we can rewrite the entire expression:
√336x^12/√7x^4 = (4x^3√21)/(√7 x^2) = 4x√(21/7) = 4x√3