Add ^3√22 + 3√22 + √22.
To add ^3√22 + 3√22 + √22, we can simplify the problem by combining like terms.
First, let's combine the terms with the same radical (^3√22 and 3√22):
^3√22 + 3√22 = (^3√22 + 3√22)
Since the two terms have the same radical (^3√22), we can add their coefficients:
(^3√22 + 3√22) = (1 + 3)√22
Simplifying further:
(1 + 3)√22 = 4√22
Now, we can add the remaining term (√22):
4√22 + √22 = 5√22
Therefore, ^3√22 + 3√22 + √22 simplifies to 5√22.
^3√22 + 3√22 + √22 can be simplified as follows:
^3√22 is the same as 22^(1/3).
3√22 is the same as 3 * 22^(1/2).
√22 is the same as 1 * 22^(1/2).
To add these terms, we need to have the same exponent for the base 22. Since ^3√22 has an exponent of 1/3, we need to convert the other terms to the same exponent. We can do this by rewriting 22^(1/2) as 22^(2/6) and 22^(1/3) as 22^(2/6).
So, ^3√22 + 3√22 + √22 becomes:
22^(1/3) + 3 * 22^(2/6) + 1 * 22^(2/6).
Now that they share the same exponent, we can add the coefficients of the terms:
1/3 + 3 * 2/6 + 2/6.
Simplifying the coefficients, we get:
1/3 + 1 + 1/3.
Combining the coefficients, we have:
1/3 + 3/3 + 1/3 = 5/3.
Therefore, ^3√22 + 3√22 + √22 equals 5/3.
To simplify the expression ^3√22 + 3√22 + √22, we can start by combining the like terms. In this case, all the terms have the same radical, which is √22.
Since there are three terms with √22, we can add the coefficients in front of the radical:
1 + 3 + 1 = 5
So, the simplified expression is 5√22.