Solve for x in 50x^(3/2) = 32x^(-1/2)?
50x^(3/2) = 32x^(-1/2)
50x^(3/2) - 32x^(-1/2) = 0
x^(-1/2) (50x^2 - 32) = 0
1/√x = 0 , no such solution
or x^2 = 32/50 = 16/25
x = ± 4/5
Pls I dont understand the solution
To solve for x in the equation 50x^(3/2) = 32x^(-1/2), we can start by simplifying the equation by getting rid of the exponent of x.
First, let's rewrite x^(-1/2) as 1/sqrt(x) (since x^(-1/2) is equivalent to 1/sqrt(x)).
The updated equation becomes: 50x^(3/2) = 32(1/sqrt(x)).
Next, let's multiply both sides of the equation by sqrt(x) to cancel out the denominators on the right side of the equation.
50x^(3/2) * sqrt(x) = 32(1/sqrt(x)) * sqrt(x).
Simplifying further, we have:
50x^(3/2) * sqrt(x) = 32.
To continue solving the equation, we can rewrite x^(3/2) as (sqrt(x))^3.
So, the equation becomes:
50(sqrt(x))^3 * sqrt(x) = 32.
Now, we can further simplify the equation by multiplying the terms inside the parentheses:
50x^(3/2) * sqrt(x) = 32.
Therefore, 50(sqrt(x))^4 = 32.
Now, divide both sides of the equation by 50 to isolate sqrt(x):
(sqrt(x))^4 = 32/50.
Simplifying further:
(sqrt(x))^4 = 16/25.
Taking the fourth root of both sides:
sqrt(x) = ± (16/25)^(1/4).
Now, we can solve for sqrt(x) by finding the fourth root of (16/25):
sqrt(x) = ± √(16/25).
Evaluating the square root:
sqrt(x) = ± 4/5.
To find the possible values of x, we square both sides:
x = (± 4/5)^2.
Simplifying further, we have two solutions:
x = 16/25 or x = 16/25.
Thus, the solutions for x are x = 16/25.
To solve for x in the equation 50x^(3/2) = 32x^(-1/2), we can begin by getting rid of the fractional exponents.
Step 1: Rewrite the equation without the fractional exponents:
50x^(3/2) = 32x^(-1/2)
can be rewritten as:
50√(x^3) = 32 / √(x)
Step 2: Simplify the square roots:
50x^(3/2) = 32 / √(x)
can be rewritten as:
50√(x^3) = 32 / (1 / √(x))
which simplifies to:
50√(x^3) = 32√(x) (since 1 / √(x) is equal to √(x))
Step 3: Simplify the equation further:
50√(x^3) = 32√(x)
Raising both sides of the equation to the power of 2 gives:
(50√(x^3))^2 = (32√(x))^2
which simplifies to:
2500x^3 = 1024x
Step 4: Move all terms to one side of the equation:
2500x^3 - 1024x = 0
Step 5: Factor out x from the equation:
x(2500x^2 - 1024) = 0
Step 6: Solve for x:
Setting each factor equal to zero, we have two possible solutions:
x = 0 or 2500x^2 - 1024 = 0
First, let's solve for x = 0:
When x = 0, the equation becomes:
50(0)^(3/2) = 32(0)^(-1/2)
This simplifies to:
0 = 0
Therefore, x = 0 is a valid solution.
Now, let's solve for 2500x^2 - 1024 = 0:
2500x^2 - 1024 = 0
This is a quadratic equation in terms of x^2. We can solve it by factoring or by using the quadratic formula.
By factoring:
Rearranging the equation, we have:
2500x^2 = 1024
Dividing both sides by 2500:
x^2 = 1024/2500
Taking the square root of both sides:
x = ±√(1024/2500)
By using the quadratic formula:
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
In our case, a = 2500, b = 0, and c = -1024. Substituting these values into the formula:
x = (-0 ± √(0^2 - 4(2500)(-1024))) / (2(2500))
Simplifying the equation:
x = ± √(1024/2500)
Therefore, the solutions are:
x = 0
x = ± √(1024/2500)
To summarize:
The solutions to the equation 50x^(3/2) = 32x^(-1/2) are x = 0 and x = ± √(1024/2500).