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Question Find the sum of 3\sqrt{3} and 3123\sqrt{12} and determine if the result is rational or irrational.

Studdy Solution

STEP 1

Assumptions1. We are given two terms 3\sqrt{3} and 3123 \sqrt{12}. . We need to find the sum of these two terms.
3. We need to simplify the result as much as possible.
4. We also need to determine whether the result is rational or irrational.

STEP 2

First, we need to simplify the second term 12 \sqrt{12}. We can do this by factoring12 into4 and, which are both perfect squares.
12=4× \sqrt{12} = \sqrt{4 \times}

STEP 3

Now, we can simplify the square root of.
312=3×233 \sqrt{12} =3 \times2 \sqrt{3}

STEP 4

Calculate the simplified form of 3123 \sqrt{12}.
312=3×23=633 \sqrt{12} =3 \times2 \sqrt{3} =6 \sqrt{3}

STEP 5

Now that we have simplified the second term, we can add the two terms together.
3+3\sqrt{3} + \sqrt{3}

STEP 6

Combine like terms.
3+63=3\sqrt{3} +6 \sqrt{3} = \sqrt{3}

STEP 7

Now we need to determine whether the result is rational or irrational. A number is rational if it can be written as the ratio of two integers, and its decimal expansion either terminates or repeats. The square root of3 is an irrational number because it cannot be written as the ratio of two integers, and its decimal expansion does not terminate or repeat. Therefore, any number multiplied by the square root of3 is also irrational.
The sum of 3\sqrt{3} and 3123 \sqrt{12} in simplest form is 737 \sqrt{3}, and this number is irrational.

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