Final answer:
To find the exact value of x in the equation 27^(-1/3) = 9^(1/4) ÷ 3^(x+1), we need to solve the equation step by step. By simplifying the equation and equating the denominators, we find that x = -1.
Step-by-step explanation:
To find the exact value of x, we need to solve the equation 27-1/3 = 91/4 ÷ 3x+1. Let's solve this step by step:
- First, let's simplify the left side of the equation. 27-1/3 is the same as taking the cube root of 27 and then finding its inverse. The cube root of 27 is 3, and its inverse is 1/3. So, 27-1/3 = 1/3.
- Now, let's simplify the right side of the equation. 91/4 is the same as taking the fourth root of 9. The fourth root of 9 is 3. So, 91/4 = 3.
- Now, we have 1/3 = 3 ÷ 3x+1.
- Let's simplify the right side of the equation further. 3 ÷ 3x+1 is the same as 3 × (3x+1)-1. Using the rule of negative exponents, (3x+1)-1 becomes 1 / (3x+1).
- So, now we have 1/3 = 1 / (3x+1).
- To find the exact value of x, we can equate the denominators of the fractions. The denominators are the same, so we can set the numerators equal: 1 = 3x+1.
- To solve for x, we can take the logarithm of both sides. Let's take the logarithm base 3 of both sides: log3(1) = log3(3x+1).
- Since log3(1) = 0, we have 0 = (x + 1) × log3(3).
- Since log3(3) = 1, we have 0 = x + 1. Solving for x, x = -1.
Therefore, the exact value of x is -1.