# Lesson Video: Cubes of Numbers Mathematics

In this video, we will learn how to determine the cube of an integer and use it to represent the volume of a cube.

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### Video Transcript

In this video, we will learn how to determine the cube of an integer and use it to represent the volume of a cube. We will begin by recalling how we square numbers and then extend this to enable us to cube numbers. We recall that in order to square a number, we multiply the number by itself. The notation for this is a superscript two. We read this expression as 𝑎 squared. This is equal to 𝑎 multiplied by 𝑎. The first 10 square numbers can be calculated by squaring the positive integers from one to 10. One squared is equal to one multiplied by one, which gives us an answer of one. Two squared is equal to two multiplied by two, which equals four, three squared is equal to three multiplied by three, which is nine, and so on. The first 10 square numbers are one, four, nine, 16, 25, 36, 49, 64, 81, and 100.

Let’s now consider what happens when we want to cube a number. To cube a number, we multiply the number by itself and itself again. This means that 𝑎 cubed denoted with a superscript three is equal to 𝑎 multiplied by 𝑎 multiplied by 𝑎. We will now look at some questions where we need to cube integers.

Evaluate two raised to the power of three.

Two raised to the power of three is written two with a superscript three after it. We know that raising any number to the power three means cubing it and that cubing any number involves multiplying the number by itself then itself again. Two cubed is equal to two multiplied by two multiplied by two. As multiplication is commutative and associative, we can multiply these numbers in any order. But for this example, we will work from left to right. Two multiplied by two is equal to four, so we’re left with four multiplied by two. As four multiplied by two is equal to eight, we know that two cubed or two raised to the power of three is equal to eight.

Our next question will involve cubing a different number.

Evaluate four cubed.

The superscript three in this question means that we need to cube the number four. When cubing any number, we need to multiply it by itself then itself again. We can perform the calculation in any order, but we will begin by multiplying the first two fours. We know that four multiplied by four is equal to 16, so four cubed is equal to 16 multiplied by four. One way of working this out is using column multiplication. Four multiplied by six is equal to 24, so we put a four in the ones column and carry the two. Four multiplied by one is four, and adding the two that we carried gives us six. 16 multiplied by four is equal to 64. This means that four cubed is also equal to 64.

In our next question, we will solve a word problem.

The number of calories in one croissant can be expressed as six cubed. Determine the whole number that represents six cubed.

We know that when we raise a number to the power of three, we want to cube the number. This involves multiplying the number by itself and itself again. Six cubed is equal to six multiplied by six multiplied by six. We know that six multiplied by six is 36, so we need to multiply this by six. One way of doing this is using the grid or box method. We split 36 into its tens and ones, so we have 30 and six. As three multiplied by six is 18, 30 multiplied by six is 180. We know that six multiplied by six is 36.

We can then find the sum of 180 and 36 using column addition. Zero plus six is equal to six. Eight plus three is equal to 11, so we need to carry the one. This leaves us with an answer of 216. The whole number that represents six cubed is therefore 216. This means that there are 216 calories in one croissant.

In our next question, we need to find the cube of a negative number.

Which of the following is equal to negative 10 cubed? Is it (A) negative 100, (B) negative 1,000, (C) 1,000, (D) one one thousandth, or (E) negative one one thousandth.

We begin by recalling that raising a number to the power of three is the same as cubing that number. When cubing any number, we multiply it by itself and itself again. This means that to calculate negative 10 cubed, we need to multiply negative 10 by negative 10 by negative 10. When multiplying two positive numbers or two negative numbers, we get a positive answer. However, when we multiply a negative by a positive or positive by a negative, we get a negative answer.

In this question, we begin by multiplying negative 10 by negative 10. As there are two negatives, our answer will be positive, and 10 multiplied by 10 is 100. We now need to multiply 100 by negative 10. We’re multiplying a positive number by a negative number. This will give us a negative answer. And as 100 multiplied by 10 is 1,000, 100 multiplied by negative 10 is negative 1,000. This means that the correct answer is option (B). Negative 1,000 is equal to negative 10 cubed.

Before answering one final question, we will look at how we can cube integers to help us calculate the volume of shapes. We know that the volume of any three-dimensional shape is the space inside it, and it is measured in cubic units, for example, cubic centimeters or cubic meters. A cube is a three-dimensional shape, where the length of each vertex or side is the same. If we let the length of each vertex or side be 𝑠 units, then the volume of the cube will be equal to the length multiplied by the width multiplied by the height, in other words, 𝑠 multiplied by 𝑠 multiplied by 𝑠. Using our knowledge of cube numbers, we know that we can write this as 𝑠 cubed. The volume of any cube is equal to the side length cubed.

We will now look at a question where we need to calculate the volume of a cube.

The amount of wax this candle mold can hold is measured in cubic units. Find how much wax it can hold by using the expression 𝑠 multiplied by 𝑠 multiplied by 𝑠, where 𝑠 is the length of a side.

We know that the volume of any 3D shape is measured in cubic units. And the volume of a cube is equal to the side length multiplied by the side length multiplied by the side length. This can also be written 𝑠 cubed. In this question, we are told that the mold has a side length of 14 units. Therefore, its volume will be equal to 14 multiplied by 14 multiplied by 14. We can begin by multiplying 14 by 14. We know that 14 multiplied by 10 is 140, and 14 multiplied by four is 56. This means that 14 multiplied by 14 or 14 squared is equal to 196.

There are lots of ways we could multiply this by 14. One way would be to multiply 200 by 14 first. This is equal to 2,800 as two multiplied by 14 is 28. We already know that four multiplied by 14 is equal to 56. We can then subtract these so that 196 multiplied by 14 is equal to 2,744. We can therefore conclude that the candle mold can hold 2,744 cubic units of wax.

We will now summarize the key points from this video. In order to cube a number, we multiply it by itself and then itself again. For example, five cubed is equal to five multiplied by five multiplied by five. This is equal to 125. The cube numbers are the answers we get when we cube the positive integers. This means that the first five are one, eight, 27, 64, and 125. These are the answers to one cubed, two cubed, three cubed, four cubed, and five cubed.

Whilst we’ve only focused on cubing integers in this video, we can use the same method to keep decimals and fractions. We also saw that the volume of any cube is equal to its side length cubed, and the units for volume are cubic units. This means that the volume of a cube with side length three centimeters is 27 cubic centimeters as three multiplied by three multiplied by three is 27.