Add 22−−√3 + 322−−√ + 22−−√ .
The expression 22−−√3 + 322−−√ + 22−−√ can be simplified as follows:
22−−√3 = 4√3
322−−√ = 9√2
22−−√ = 2√2
Therefore, 22−−√3 + 322−−√ + 22−−√ = 4√3 + 9√2 + 2√2.
Combining like terms, we get:
4√3 + 9√2 + 2√2 = 4√3 + (9+2)√2 = 4√3 + 11√2.
Hence, the simplified expression is 4√3 + 11√2.
To add these numbers, we need to simplify each square root individually and then add them together. Let's start with simplifying each square root:
√3 can't be simplified any further because 3 is prime.
√32 can be simplified as √16 * √2. Since √16 = 4, we can rewrite it as 4√2.
√2 can't be simplified any further.
Now let's add them together:
22−−√3 + 322−−√ + 22−−√ = 2√3 + 4√2 + 2√2
To add the like terms with the same radical (√3 and √2), we combine their coefficients:
2√3 + 4√2 + 2√2 = 2√3 + (4 + 2)√2 = 2√3 + 6√2
So, the simplified sum is 2√3 + 6√2.
To add these numbers with square roots, we need to simplify each square root term first. Let's begin by simplifying each term individually:
Term 1: √3
The square root of 3 cannot be simplified any further, so we leave it as √3.
Term 2: 32−√
To simplify √32, we need to find the largest perfect square that can be evenly divided into 32. In this case, it is 16 (which is 4^2). So we can simplify √32 as 4√2.
Term 3: √2
Since the square root of 2 cannot be simplified any further, we leave it as √2.
Now, let's rewrite the expression using the simplified terms:
√3 + 4√2 + √2
To add these terms, we combine like terms. In this case, we have two terms with √2:
(√3 + √2) + 4√2
Adding √3 and √2 gives us:
√3 + √2 = √3 + √2
Finally, combining this with 4√2:
(√3 + √2) + 4√2 = √3 + √2 + 4√2
So, the simplified expression is √3 + √2 + 4√2.