Grade 9Mathematics

Cubes and Cube Roots

Cubes and cube roots; estimating; using mathematical tables; applications.

📖 4 min read · 3 worked examples · 8 practice questions

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The lesson

First, a cube means raising a number to the third power, written n³. That's the same as multiplying the number by itself three times: n × n × n. The cube root, written ∛n, asks the opposite question: which number multiplied by itself three times gives us n? Why does this matter? In geometry, the volume of a rectangular prism is length × width × height – exactly a cube of a length when all sides are equal. Knowing cube roots lets us find the side length of a cubic container when we know its volume. For example, imagine a water tank that holds 1,000 cubic metres of water. The cube root of 1,000 is 10, so the tank could be a perfect cube 10 m on each side. Or think about soil blocks used by farmers: if a block's volume is 27 m³, the side length is ∛27 = 3 m. These everyday links show why cubes and cube roots are useful. To recap: a cube is n³, the cube root ∛n reverses it, and both help us calculate and understand volumes in real life. Any questions before we move on?

Everyone, let's explore the properties of cubes. We'll see how quickly they grow as the base increases. First, notice this pattern: each time we add 1 to the base, the cube value is roughly three times larger than the previous one. For example, 2³ is 8, and 3³ jumps to 27—about three times bigger. At the bar chart. You can see the bars rising steeply from 1³ up to 10³, illustrating just how rapidly the numbers expand. A second key point: the difference between neighboring cubes grows fast. The gap between 4³ (64) and 5³ (125) is 61, much larger than the gap between 1³ and 2³. Remember these patterns—they help you estimate cube values and make rough cube‑root calculations much easier.

Let's wrap up with a quick recap and reflection on what we've learned about cubes and cube roots. First, remember that cubes grow rapidly—each time you increase the side length by one unit, the volume multiplies by the cube of that number. Conversely, taking a cube root lets us reverse that growth and find the original side length. A handy estimation trick is to look at the neighbouring cubes. If you need the cube root of a number between 64 (4³) and 125 (5³), you can quickly guess a value between 4 and 5 and refine from there. Finally, consider real‑world uses: calculating the volume of a grain silo, determining how much material is needed for a concrete block, or figuring out trade quantities when goods are packed in cubic containers. Take a moment to think about where you might have seen cubes in everyday life, and how estimating their dimensions could help you solve practical problems.

Worked examples

Worked Example 1

Everyone, let's dive into Worked Example 1, where we'll calculate 4 cubed and see what that number means. First, we compute 4 × 4 × 4. Multiplying the first two fours gives 16, and 16 × 4 equals 64. Let's quickly verify that result on the calculator. Type 4 ^ 3 or press 4 × 4 × 4 – you should see 64 appear. Finally, think of that 64 as the volume of a cube whose sides are each 4 metres long. Imagine a wooden crate 4 m on each edge – its capacity is 64 cubic metres. We've computed the cube, confirmed it with a calculator, and related the number to a real‑world volume. Any questions before we move on?

– Estimating a Cube Root

Let's work through Example 2, where we estimate the cube root of 200. First, we look for the nearest perfect cubes. We know 5³ equals 125 and 6³ equals 216, so the true cube root must lie between 5 and 6. Next, we narrow it down using a cube‑root table. The table shows that ∛200 is about 5.85, giving us a good estimate. Notice how the estimate falls nicely between the two known cubes—just as we expected. Any questions before we move on?

– Application in Agriculture

Let's work through Example 3, which shows how we can size a cubic storage box for a farm's water supply. First, we convert the given volume: 5000 litres is the same as 5 cubic metres, because one cubic metre equals 1000 litres. Next, we need the length of each side of a cube whose volume is 5 m³. That means we take the cube root of 5, written as ∛5, which is about 1.71 metres. The result is that every side of the storage box should be roughly 1.71 metres long – just enough to hold 5000 litres of water. Any questions before we move on to the next example?

Practice questions

First, think about what it means to raise a number to the third power. 3³ means multiplying three by itself three times: 3 × 3 × 3, which equals 27.
When we talk about the volume of a cube, we also multiply the side length three times—length × width × height. With a side of 3 cm, the volume is 3 cm × 3 cm × 3 cm = 27 cubic centimeters.
If you chose the right answers, you've demonstrated a solid understanding of exponentiation and how it applies to real‑world measurements. If you missed any, review the steps we just walked through, and you'll have it down next time.
First, look at the list of known cubes. Since 5³ = 125 and 6³ = 216, we know ∛150 must fall between 5 and 6.
Use the detailed table to hone your estimate. Notice that 5.
First problem: Find the cube of 7. Remember, cubing means multiplying the number by itself three times.
Second, we need to estimate the cube root of 400. Think about which perfect cubes are close to 400 and use interpolation.
Third, a cubic crate holds 27 cubic metres. What is the length of each side?

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Cubes and Cube Roots

📖 The lesson

✏️ Worked examples