If $\left|\overrightarrow{a}\right|=2$ and $\left|\overrightarrow{b}\right|=3$ and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60°$, then $3|3\overrightarrow{a}+4\overrightarrow{b}|+4|3\overrightarrow{a}-4\overrightarrow{b}|$ is equal to ___.

The correct answer is:

$18\sqrt{7}+24\sqrt{3}$

Step-by-step Explanation:

We are given the magnitudes of two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, and the angle between them:
$\left|\overrightarrow{a}\right|=2$
$\left|\overrightarrow{b}\right|=3$
The angle $\theta$ between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60°$.

First, let's find the dot product of the two vectors:
$\overrightarrow{a}·\overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta$
Substituting the given values:
$\overrightarrow{a}·\overrightarrow{b}=2·3·\cos 60°=6·\frac{1}{2}=3$

Next, let's calculate the magnitude squared of the vector $3\overrightarrow{a}+4\overrightarrow{b}$:
$|3\overrightarrow{a}+4\overrightarrow{b}{|}^2=9|\overrightarrow{a}{|}^2+16|\overrightarrow{b}{|}^2+24(\overrightarrow{a}·\overrightarrow{b})$
Substitute the magnitudes and the dot product:
$|3\overrightarrow{a}+4\overrightarrow{b}{|}^2=9(2^2)+16(3^2)+24(3)$
$|3\overrightarrow{a}+4\overrightarrow{b}{|}^2=9(4)+16(9)+72$
$|3\overrightarrow{a}+4\overrightarrow{b}{|}^2=36+144+72=252$
Taking the square root:
$|3\overrightarrow{a}+4\overrightarrow{b}|=\sqrt{252}=\sqrt{36·7}=6\sqrt{7}$

Now, let's calculate the magnitude squared of the vector $3\overrightarrow{a}-4\overrightarrow{b}$:
$|3\overrightarrow{a}-4\overrightarrow{b}{|}^2=9|\overrightarrow{a}{|}^2+16|\overrightarrow{b}{|}^2-24(\overrightarrow{a}·\overrightarrow{b})$
Substitute the values:
$|3\overrightarrow{a}-4\overrightarrow{b}{|}^2=36+144-72=108$
Taking the square root:
$|3\overrightarrow{a}-4\overrightarrow{b}|=\sqrt{108}=\sqrt{36·3}=6\sqrt{3}$

Finally, we compute the value of the required expression:
$3|3\overrightarrow{a}+4\overrightarrow{b}|+4|3\overrightarrow{a}-4\overrightarrow{b}|$
Substituting our derived values:
$=3\left(6\sqrt{7}\right)+4\left(6\sqrt{3}\right)$
$=18\sqrt{7}+24\sqrt{3}$