Let's break down the problem step-by-step to find the sum of [tex]\( 4\sqrt{9} \)[/tex] and [tex]\( 3\sqrt{10} \)[/tex], and then determine whether the result is rational or irrational.
### Step 1: Calculate [tex]\( 4\sqrt{9} \)[/tex]
First, we calculate [tex]\( \sqrt{9} \)[/tex]:
[tex]\[ \sqrt{9} = 3 \][/tex]
Next, multiply by 4:
[tex]\[ 4\sqrt{9} = 4 \cdot 3 = 12 \][/tex]
### Step 2: Calculate [tex]\( 3\sqrt{10} \)[/tex]
Since [tex]\( \sqrt{10} \)[/tex] is already in its simplest radical form and is an irrational number, we can write:
[tex]\[ 3\sqrt{10} = 3 \cdot \sqrt{10} \][/tex]
### Step 3: Sum the Two Terms
Now we will sum [tex]\( 4\sqrt{9} \)[/tex] and [tex]\( 3\sqrt{10} \)[/tex]:
[tex]\[ 4\sqrt{9} + 3\sqrt{10} = 12 + 3\sqrt{10} \][/tex]
So the result in its simplest form is:
[tex]\[ 12 + 3\sqrt{10} \][/tex]
### Step 4: Determine Whether the Result is Rational or Irrational
To determine if the result is rational or irrational, recall the definitions:
- A rational number can be expressed as the quotient or fraction [tex]\(\frac{p}{q}\)[/tex] of two integers [tex]\(p\)[/tex] and [tex]\(q\)[/tex], with the denominator [tex]\(q\)[/tex] not equal to zero.
- An irrational number cannot be expressed as such a fraction and its decimal expansion is non-terminating and non-repeating.
In [tex]\( 12 + 3\sqrt{10} \)[/tex]:
- 12 is a rational number.
- [tex]\( 3\sqrt{10} \)[/tex] is an irrational number, because [tex]\(\sqrt{10}\)[/tex] is irrational and multiplying an irrational number by a non-zero rational number (in this case, 3) results in an irrational number.
The sum of a rational number (12) and an irrational number ([tex]\( 3\sqrt{10} \)[/tex]) is always irrational.
### Conclusion
The result, [tex]\( 12 + 3\sqrt{10} \)[/tex], is an irrational number. This is because it includes the term [tex]\( 3\sqrt{10} \)[/tex], which is irrational, and the sum of a rational and an irrational number is irrational.
