Certainly! Let's rationalize the denominator of [tex]\(\frac{14}{\sqrt{7}}\)[/tex] and simplify the expression step by step.
### Step 1: Identify the need to rationalize the denominator
The given expression is [tex]\(\frac{14}{\sqrt{7}}\)[/tex]. The denominator has a square root, which we need to eliminate to rationalize it.
### Step 2: Multiply by a form of 1 to eliminate the square root in the denominator
To rationalize the denominator, multiply both the numerator and the denominator by [tex]\(\sqrt{7}\)[/tex] (since multiplying by [tex]\(\frac{\sqrt{7}}{\sqrt{7}}\)[/tex] is equivalent to multiplying by 1 and does not change the value of the expression).
So, we have:
[tex]\[ \frac{14}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{14 \cdot \sqrt{7}}{\sqrt{7} \cdot \sqrt{7}} \][/tex]
### Step 3: Simplify the denominator
The denominator [tex]\(\sqrt{7} \cdot \sqrt{7}\)[/tex] is [tex]\(7\)[/tex], because the product of a square root with itself equals the number inside the square root:
[tex]\[ \sqrt{7} \cdot \sqrt{7} = 7 \][/tex]
Therefore, our expression simplifies to:
[tex]\[ \frac{14 \cdot \sqrt{7}}{7} \][/tex]
### Step 4: Simplify the entire expression
Now, divide the numerator by the denominator:
[tex]\[ \frac{14 \cdot \sqrt{7}}{7} = 2 \cdot \sqrt{7} \][/tex]
### Step 5: Get the numerical approximation
Finally, if we want to express the answer as a numerical approximation, we can compute [tex]\(\sqrt{7} \approx 2.6457513110645906\)[/tex].
Thus:
[tex]\[ 2 \cdot \sqrt{7} \approx 2 \cdot 2.6457513110645906 = 5.291502622129181 \][/tex]
So, the rationalized form of [tex]\(\frac{14}{\sqrt{7}}\)[/tex] is [tex]\(2\sqrt{7}\)[/tex], and its numerical value is approximately [tex]\(5.291502622129181\)[/tex].
This completes the rationalization and simplification of [tex]\(\frac{14}{\sqrt{7}}\)[/tex].
