What is the value of the expression when $n=3$ ?
$\frac{6\left(n^2+2\right)}{n}$
We are given the mathematical expression \(\frac{6(n^2+2)}{n}\) and we need to evaluate it for the specific value \(n=3\). Our objective is to substitute \(n=3\) into the expression, simplify the calculation according to the order of operations, and identify the correct result among the provided options: 16, 22, 30, and 66.
To evaluate the expression \(\frac{6(n^2+2)}{n}\) for \(n=3\), we first substitute 3 for \(n\):
\[\frac{6(3^2+2)}{3}\]
Next, we perform the operations inside the parentheses. First, calculate the square of 3:
\[3^2 = 9\]
Then, add 2 to the result:
\[9 + 2 = 11\]
Now the expression simplifies to:
\[\frac{6(11)}{3}\]
Multiply 6 by 11:
\[6 \times 11 = 66\]
Finally, divide by 3:
\[\frac{66}{3} = 22\]
Thus, the value of the expression is 22.
By evaluating the given expression \(\frac{6(n^2+2)}{n}\) at \(n=3\), we have calculated the result to be 22. Comparing this result with the provided options, we see that 22 is one of the choices. Therefore, the correct value for the expression is 22.
In real life, evaluating algebraic expressions is crucial for budgeting and planning. For example, if you are planning an event where the total cost depends on the number of guests \(n\), you might use a formula like this one to estimate expenses. If the formula for the total cost in hundreds of dollars is \(\frac{6(n^2+2)}{n}\), substituting \(n=3\) helps you calculate the cost for three groups or sections accurately, ensuring you stay within your budget.