Video: Finding the Volume of a Cube given Its Edge Length

Find the volume of the cube, given that the length of its edge is (1/10) ∛12 cm.

03:10

Video Transcript

Find the volume of the cube given that the length of its edge is one-tenth of the cubed root of 12 centimeters.

So as we can see from the question, each of the edges of our cube are a tenth of the cube root of 12 centimeters long. And I’ve labelled this on the diagram. Well, I’ve shown three of the lengths. So we’ve got one that might be considered a width, one that might be considered depth, and one might be considered a height. In fact, they’re all just lengths of our cube. And the reason I’ve done this is because the volume of a cube is equal to 𝐿 cubed, the length of the cube cubed.

So therefore using this formula, we can say that the volume is gonna be equal to a tenth of the cube root of 12, and then this is all cubed, and then we can rewrite this using one of our index laws. And that law is that the cube root of 𝑎 is equal to 𝑎 to the power of third. And when we use this rule, we can rewrite our equation as the volume is equal to 12 to the power of a third over 10 all to the power of three. And then we can use another rule to simplify even further. And that is that if we have 𝑎 over 𝑏 all to the power of 𝑐, this is equal to 𝑎 to the power of 𝑐 divided by 𝑏 to the power 𝑐.

So then using this rule, we can say that the volume is equal to 12 to the power of a third, and then this is all cubed, divided by 10 cubed. And then we can use one final rule to help us to simplify even further. And that is a rule that tells us that if we have 𝑎 to power of 𝑏 all to the power of 𝑐, this is equal to 𝑎 to the power 𝑏𝑐 or 𝑏 multiplied by 𝑐. So when we do that and use this rule, we get the volume is equal to 12. And it’s 12 because we had 12 to the power of a third all cubed. Well we multiply the powers together, so we do three multiplied by a third, which gives us one, so 12 to power of one or just 12. And this is over a 1000, and that’s because 10 to the power of three of 10 cubed is 1000 because 10 multiplied by 10 is 100 multiplied by 10 again is 1000.

So therefore, we can say that the volume is an equal to 0.012 centimeters cubed. And we got that because we had 12 divided by a 1000. If you’re dividing by 1000, we move each digit three place values to the right. So that would be a decimal answer. If we wanted a fraction answer, we will divide the numerator and the denominator by four. That’s because four is a common factor for both of those. And when we do that, we can say that the volume is equal to three over 250 centimeters cubed. So therefore, we can say that if we want to find the volume of a cube whose length of edge is one over 10 or one-tenth of the cube root of 12 centimeters, then the volume is going to be equal to 0.012 centimeters cubed or three over 250 centimeters cubed.

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