solve
1:
n^3/2 = 125
2:
(x - 12)^3/2 = 64
3:
3125 = (-1-18t)^5/3
I will do the second, you do the others in the same way.
(x - 12)^3/2 = 64 , raise both sides to the 2/3, (notice the reciprocal of 3/2)
[(x - 12)^3/2]^(2/3) = 64^(2/3)
x-12 = 16
x = 28
1. n^(3/2) = 125.
Take cube root of both sides(raise to 1/3 power):
n^(1/2) = 5,
Sqrt(n) = 5,
Square both sides:
n = 25.
check: 25^(3/2) = 25 * 25^(1/2) = 25 * sqrt(25) = 25 * 5 = 125.
Sure! Let's solve these equations step by step.
1. Solving the equation n^(3/2) = 125:
To find the value of n, we can start by raising both sides of the equation to the reciprocal of the exponent 3/2, which is 2/3. This will give us:
(n^(3/2))^(2/3) = 125^(2/3)
Applying the property of exponents, we simplify the left side of the equation:
n^((3/2)*(2/3)) = 125^(2/3)
n^1 = 125^(2/3)
Simplifying further:
n = 125^(2/3)
Now we need to calculate 125^(2/3). This can be done by applying exponentiation rules:
125^(2/3) = (5^3)^(2/3)
125^(2/3) = 5^(3*(2/3))
125^(2/3) = 5^2
125^(2/3) = 25
Therefore, the solution to the equation n^(3/2) = 125 is:
n = 25
2. Solving the equation (x - 12)^(3/2) = 64:
To find the value of x, we can raise both sides of the equation to the reciprocal of the exponent 3/2, which is 2/3. This will give us:
((x - 12)^(3/2))^(2/3) = 64^(2/3)
Simplifying the left side of the equation:
(x - 12)^((3/2)*(2/3)) = 64^(2/3)
(x - 12)^1 = 64^(2/3)
Simplifying further:
x - 12 = 64^(2/3)
Now, we need to calculate 64^(2/3) using exponentiation rules as mentioned earlier:
64^(2/3) = (4^3)^(2/3)
64^(2/3) = 4^(3*(2/3))
64^(2/3) = 4^2
64^(2/3) = 16
Adding 12 to both sides of the equation, we get:
x - 12 + 12 = 16 + 12
x = 28
Therefore, the solution to the equation (x - 12)^(3/2) = 64 is:
x = 28
3. Solving the equation 3125 = (-1 - 18t)^(5/3):
To find the value of t, we can start by raising both sides of the equation to the reciprocal of the exponent 5/3, which is 3/5. This will give us:
(3125)^(3/5) = ((-1 - 18t)^(5/3))^(3/5)
Simplifying the left side of the equation:
3125^(3/5) = (-1 - 18t)^((5/3)*(3/5))
3125^(3/5) = (-1 - 18t)^1
3125^(3/5) = -1 - 18t
Now, let's solve for t. Subtracting 1 from both sides of the equation:
3125^(3/5) - 1 = -1 - 18t - 1
3125^(3/5) - 1 = -18t - 2
To isolate t, divide both sides of the equation by -18:
(3125^(3/5) - 1) / -18 = (-18t - 2) / -18
[(3125^(3/5) - 1) / -18] * (-1/18) = t
Therefore, the solution to the equation 3125 = (-1 - 18t)^(5-2/3) is:
t = [(3125^(3/5) - 1) / -18] * (-1/18)