solve the following equation with a rational exponent 2x 3/2 = 16
To solve the equation 2x^(3/2) = 16, we can rewrite it as:
x^(3/2) = 16/2
x^(3/2) = 8
Now, we will cube both sides of the equation to eliminate the rational exponent of 3/2:
(x^(3/2))^3 = 8^3
x^(9/2) = 512
Now, we can rewrite the equation with a rational exponent as:
x^(9/2) - 512 = 0
To solve this equation, we can raise both sides of the equation to the power of 2/9:
(x^(9/2))^(2/9) - 512^(2/9) = 0^(2/9)
x^(9/2 * 2/9) - 512^(2/9) = 0
x^(1/1) - 2 = 0
x - 2 = 0
x = 2
Therefore, the solution to the equation 2x^(3/2) = 16 is x = 2.
solve the following equation with a rational exponent: 5x 4/3 = 80
To solve the equation 5x^(4/3) = 80, we can rewrite it as:
x^(4/3) = 80/5
x^(4/3) = 16
Now, we will raise both sides of the equation to the power of 3/4 to eliminate the rational exponent of 4/3:
(x^(4/3))^(3/4) = 16^(3/4)
x^(4/3 * 3/4) = 16^(3/4)
x^(1/1) = 16^(3/4)
x = 16^(3/4)
To simplify 16^(3/4), we can rewrite it as (2^4)^(3/4) since 16 is 2 raised to the power of 4:
x = (2^4)^(3/4)
x = 2^(4 * 3/4)
x = 2^3
x = 8
Therefore, the solution to the equation 5x^(4/3) = 80 is x = 8.
solve the following equation with a rational exponent x 2/3 = 64
To solve the equation x^(2/3) = 64, we can rewrite it as:
x = 64^(3/2)
To simplify 64^(3/2), we can rewrite it as (8^2)^(3/2) since 64 is 8 raised to the power of 2:
x = (8^2)^(3/2)
x = 8^(2 * 3/2)
x = 8^3
x = 512
Therefore, the solution to the equation x^(2/3) = 64 is x = 512.
To solve the equation 2x^(3/2) = 16, you can follow these steps:
Step 1: Divide both sides of the equation by 2 to isolate the variable term:
2x^(3/2) / 2 = 16 / 2
x^(3/2) = 8
Step 2: Rewrite the rational exponent as a radical expression:
x^(3/2) = √(x^3)
Step 3: Take the square root of both sides of the equation to eliminate the exponent 3/2:
√(x^3) = √8
Step 4: Simplify the square root of 8:
√8 = 2√2
Step 5: Rewrite the equation:
x^(3/2) = 2√2
Step 6: Raise both sides of the equation to the power of 2/3 to eliminate the exponent 3/2:
(x^(3/2))^(2/3) = (2√2)^(2/3)
Step 7: Apply the power of a power rule to the left side of the equation, and simplify the right side:
x^((3/2)(2/3)) = (2√2)^(2/3)
x^1 = (2√2)^(2/3)
Step 8: Simplify the exponents:
x = (2√2)^(2/3)
Step 9: Convert the right side to a radical expression:
x = ∛((2√2)^2)
Step 10: Square the inner term:
x = ∛(4 * 2)
Step 11: Simplify:
x = ∛8
Step 12: Evaluate the cube root of 8 to find the solution:
x = 2
Therefore, the solution to the equation 2x^(3/2) = 16 is x = 2.
To solve the equation 2x^(3/2) = 16 with a rational exponent, follow these steps:
Step 1: Rewrite the equation using the exponent rules. In this case, the base is 2 and the exponent is 3/2. Recall that x^(a/b) is equivalent to the (bth root of x)^a. Applying this rule, the equation becomes (2^(3/2))^x = 16, or (sqrt(2^3))^x = 16.
Step 2: Simplify the expression. The square root of 2^3 is the same as the square root of 8, which simplifies to 2*sqrt(2). Therefore, the equation becomes (2*sqrt(2))^x = 16.
Step 3: Isolate the base. Take the logarithm of both sides of the equation to move the x from the exponent position: log((2*sqrt(2))^x) = log(16). Using the logarithmic property log(a^b) = b*log(a), the equation becomes x*log(2*sqrt(2)) = log(16).
Step 4: Evaluate the logarithmic expressions. The value of log(2*sqrt(2)) can be calculated as log(2) + log(sqrt(2)). The natural logarithm (ln) or the base-10 logarithm (log) can be used, depending on the context or preference.
Step 5: Simplify and solve for x. Apply the logarithmic properties log(a*b) = log(a) + log(b) and log(sqrt(a)) = (1/2)*log(a):
x*(log(2) + log(sqrt(2))) = log(16),
x*(log(2) + (1/2)*log(2)) = log(16),
x*(log(2) + (1/2)*log(2)) = log(2^4),
x*(log(2) + (1/2)*log(2)) = 4*log(2),
x*(3/2)*log(2) = 4*log(2).
Now, divide both sides of the equation by (3/2)*log(2) to solve for x:
x = (4*log(2)) / ((3/2)*log(2)).
Step 6: Simplify the expression using logarithmic properties. Notice that log(2) appears on both the numerator and denominator, which cancels out:
x = 4 / (3/2),
x = 4 * (2/3),
x = 8/3.
Therefore, the solution to the equation 2x^(3/2) = 16, in terms of a rational exponent, is x = 8/3.