add ^3√22 + 3√22 + √22
Adding the terms together, we get:
√22 + √22 + √22 = 3√22
To add the given expression, let's first simplify each term and then add them together:
Step 1: Simplify ^3√22
To simplify a cube root, we need to find the largest perfect cube that divides the number.
In this case, the largest perfect cube that divides 22 is 8 (2^3).
So, ^3√22 can be written as ^3√(2^3 * 11).
Using the property ^3√(a*b) = ^3√a * ^3√b, we can rewrite ^3√(2^3 * 11) as (^3√2^3) * (^3√11).
So, ^3√22 becomes 2 * (^3√11).
Step 2: Simplify 3√22
To simplify a square root, we need to find the largest perfect square that divides the number.
In this case, the largest perfect square that divides 22 is 4 (2^2).
So, 3√22 can be written as 3√(2^2 * 11).
Using the property 3√(a*b) = 3√a * 3√b, we can rewrite 3√(2^2 * 11) as (3√2^2) * (3√11).
So, 3√22 becomes 2 * (3√11).
Step 3: Simplify √22
Since there are no perfect squares that divide 22, √22 remains as it is.
Step 4: Add the simplified terms together:
We can now add the simplified terms:
2 * (^3√11) + 2 * (3√11) + √22.
Since (^3√11) and (3√11) are like terms, we can combine them:
2 * (^3√11) + 2 * (3√11) = 2 * (^3√11 + 3√11).
Now, we can add the combined like terms with the remaining term:
2 * (^3√11 + 3√11) + √22 = 2 * 5√11 + √22.
Therefore, the simplified expression is 2 * 5√11 + √22.
To simplify this expression, we can combine the like terms. However, we need to express each radical in the same form first.
^3√22 is a cube root, while 3√22 and √22 are square roots.
To combine the like terms, we can rewrite them with the same index. We can rewrite ^3√22 as (√22)^(3/1), which is the same as ∛22.
Using this form, we can now combine the three terms:
∛22 + 3√22 + √22
We won't be able to simplify this further since they have different radicals. The final answer would be:
∛22 + 3√22 + √22