7^3/4+6^1/12=14^1/12
To solve this equation, we need to simplify each side step by step.
Let's start with the left side of the equation:
The exponent 3/4 means the cube root of 7 raised to the power of 3. Taking the cube root of 7 gives us ∛7, and raising that to the power of 3 gives (∛7)^3.
Similarly, the exponent 1/12 means the twelfth root of 6 raised to the power of 1. Taking the twelfth root of 6 gives us ∛¹²√6, and raising that to the power of 1 gives (∛¹²√6)^1.
So, the left side of the equation becomes (∛7)^3 + (∛¹²√6)^1.
Now, let's simplify the right side:
The exponent 1/12 means the twelfth root of 14 raised to the power of 1. So, the right side of the equation becomes (∛¹²√14)^1.
To check if the equation is true, we need to determine if both sides are equal.
First, simplify the left side:
(∛7)^3 + (∛¹²√6)^1 = 7^(3/4) + 6^(1/12).
Next, simplify the right side:
(∛¹²√14)^1 = 14^(1/12).
To determine if the equation is true, we need to evaluate these expressions:
Evaluating the left side, we calculate (∛7)^3 + (∛¹²√6)^1 ≈ 2.973 + 1.395 ≈ 4.368.
Evaluating the right side, we calculate (∛¹²√14)^1 ≈ 2.443.
Since 4.368 is not equal to 2.443, the equation 7^(3/4) + 6^(1/12) = 14^(1/12) is false.
To confirm whether the equation 7^(3/4) + 6^(1/12) is equal to 14^(1/12), let's break it down step by step:
Step 1: Simplify the exponents on each side of the equation:
7^(3/4) = (7^3)^(1/4) = 343^(1/4)
6^(1/12) = (6^1)^(1/12) = 6^(1/12)
Step 2: Calculate each side of the equation:
343^(1/4) = 7
6^(1/12) = 1.8171205928321397
Step 3: Add the values on the left side:
7 + 1.8171205928321397 = 8.81712059283214
Step 4: Calculate the right side of the equation:
14^(1/12) = 1.8171205928321397
Step 5: Compare the result on both sides:
8.81712059283214 is not equal to 1.8171205928321397
Therefore, the equation 7^(3/4) + 6^(1/12) does NOT equal 14^(1/12).