Question

the vault shown is in the shape of a cube. if each side is $4y^{2}$ feet, find its volume. the volume of the vault is (simplify your answer.)

Explanation:

Step1: Recall the volume formula for a cube

The volume \( V \) of a cube with side length \( s \) is given by the formula \( V = s^3 \).

Step2: Substitute the given side length into the formula

Here, the side length \( s = 4y^2 \) feet. So we need to calculate \( (4y^2)^3 \).

Step3: Apply the power of a product rule

Using the rule \( (ab)^n = a^n b^n \), we have \( (4y^2)^3 = 4^3 \times (y^2)^3 \).

Step4: Calculate the powers

We know that \( 4^3 = 64 \) and using the power of a power rule \( (a^m)^n = a^{mn} \), we get \( (y^2)^3 = y^{2\times3}=y^6 \).

Step5: Multiply the results

Multiplying these together, we have \( 4^3\times(y^2)^3 = 64\times y^6 = 64y^6 \).

Answer:

The volume of the vault is \( 64y^6 \) cubic feet.