Question

1. calculate the volume of each figure. a figure a: each side length is 2\sqrt{3}-\sqrt{2} 2\sqrt{3}-\sqrt{2}

Explanation:

Step1: Recall volume formula for cube

The volume $V$ of a cube with side - length $s$ is $V = s^3$. Here, $s=2\sqrt{3}-\sqrt{2}$.

Step2: Expand $(2\sqrt{3}-\sqrt{2})^3$ using the formula $(a - b)^3=a^3-3a^2b + 3ab^2 - b^3$

where $a = 2\sqrt{3}$ and $b=\sqrt{2}$.
First, $a^3=(2\sqrt{3})^3=2^3\times(\sqrt{3})^3=8\times3\sqrt{3}=24\sqrt{3}$.
Second, $3a^2b=3\times(2\sqrt{3})^2\times\sqrt{2}=3\times12\times\sqrt{2}=36\sqrt{2}$.
Third, $3ab^2=3\times2\sqrt{3}\times(\sqrt{2})^2=3\times2\sqrt{3}\times2 = 12\sqrt{3}$.
Fourth, $b^3=(\sqrt{2})^3 = 2\sqrt{2}$.
Then $(2\sqrt{3}-\sqrt{2})^3=24\sqrt{3}-36\sqrt{2}+12\sqrt{3}-2\sqrt{2}$.

Step3: Combine like - terms

$(2\sqrt{3}-\sqrt{2})^3=(24\sqrt{3}+12\sqrt{3})+(-36\sqrt{2}-2\sqrt{2})=36\sqrt{3}-38\sqrt{2}$.

Answer:

$36\sqrt{3}-38\sqrt{2}$