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Answer :
To find the volume of a cube when the side length [tex]\( s \)[/tex] is given, we simply use the formula for the volume of a cube, which is [tex]\( V = s^3 \)[/tex].
Here, the side length [tex]\( s \)[/tex] is given by [tex]\( 4x^2 + 3 \)[/tex]. To find the volume, we need to raise this expression to the third power:
[tex]\[
V = (4x^2 + 3)^3
\][/tex]
Now, we'll expand this expression:
1. Use the binomial expansion formula for cube [tex]\((a + b)^3\)[/tex], which is given by:
[tex]\[
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\][/tex]
Here, [tex]\( a = 4x^2 \)[/tex] and [tex]\( b = 3 \)[/tex].
2. Calculate each term:
- [tex]\( a^3 = (4x^2)^3 = 64x^6 \)[/tex]
- [tex]\( 3a^2b = 3(4x^2)^2 \cdot 3 = 3 \cdot 16x^4 \cdot 3 = 144x^4 \)[/tex]
- [tex]\( 3ab^2 = 3 \cdot 4x^2 \cdot 3^2 = 3 \cdot 4x^2 \cdot 9 = 108x^2 \)[/tex]
- [tex]\( b^3 = 3^3 = 27 \)[/tex]
3. Add all these terms together:
[tex]\[
V = 64x^6 + 144x^4 + 108x^2 + 27
\][/tex]
So, the volume of the cube is [tex]\( 64x^6 + 144x^4 + 108x^2 + 27 \)[/tex].
From the given options, the correct answer is:
[tex]\( 64x^6 + 144x^4 + 108x^2 + 27 \)[/tex].
Here, the side length [tex]\( s \)[/tex] is given by [tex]\( 4x^2 + 3 \)[/tex]. To find the volume, we need to raise this expression to the third power:
[tex]\[
V = (4x^2 + 3)^3
\][/tex]
Now, we'll expand this expression:
1. Use the binomial expansion formula for cube [tex]\((a + b)^3\)[/tex], which is given by:
[tex]\[
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\][/tex]
Here, [tex]\( a = 4x^2 \)[/tex] and [tex]\( b = 3 \)[/tex].
2. Calculate each term:
- [tex]\( a^3 = (4x^2)^3 = 64x^6 \)[/tex]
- [tex]\( 3a^2b = 3(4x^2)^2 \cdot 3 = 3 \cdot 16x^4 \cdot 3 = 144x^4 \)[/tex]
- [tex]\( 3ab^2 = 3 \cdot 4x^2 \cdot 3^2 = 3 \cdot 4x^2 \cdot 9 = 108x^2 \)[/tex]
- [tex]\( b^3 = 3^3 = 27 \)[/tex]
3. Add all these terms together:
[tex]\[
V = 64x^6 + 144x^4 + 108x^2 + 27
\][/tex]
So, the volume of the cube is [tex]\( 64x^6 + 144x^4 + 108x^2 + 27 \)[/tex].
From the given options, the correct answer is:
[tex]\( 64x^6 + 144x^4 + 108x^2 + 27 \)[/tex].
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