Hello, and welcome to today's mathematics lesson. In this session, we are going to explore Chapter Four: Cubes and Cube Roots. By the end of this lesson, you will understand what cubes and perfect cubes are, how to identify them using prime factorization, and how to find cube roots of various types of numbers.
Let us begin with the fundamental idea of a cube. For any number m, when we multiply it by itself three times — that is, m × m × m — we call this the cube of m, or m³. This is also described as m raised to the power 3.
Here are some examples to make this clear. The cube of 5 is 5³, which equals 5 × 5 × 5, giving us 125. The cube of 8 is 8³, which is 8 × 8 × 8, equal to 512. Now, what happens with negative numbers? The cube of negative 4 is (–4)³, which equals (–4) × (–4) × (–4). Since negative times negative gives positive, and then positive times negative gives negative, we get negative 64.
Notice this important pattern. The cube of a positive number is always positive. The cube of a negative number is always negative. This is different from squares, where the square of any number is always positive.
Now let us understand what we call a perfect cube, or cube number. When a number is expressed as the cube of some other number, we call it a perfect cube. For instance, since 6³ = 216, we say that 216 is a perfect cube. Similarly, 15³ = 3375, so 3375 is a perfect cube.
There is another way to recognize perfect cubes. You can recognize a perfect cube by its prime factorization: every prime factor must appear in triplets of equal factors. Let me show you what this means.
Take the number 216. When we factorize it, we get 2 × 2 × 2 × 3 × 3 × 3. We can group these as (2 × 3) multiplied by (2 × 3) multiplied by (2 × 3), which is 6 × 6 × 6, or 6³. Since 216 is made of triplets of equal factors, it is indeed a perfect cube.
Let us check whether 297 is a perfect cube. Factorizing 297, we get 3 × 3 × 3 × 11. Here, we have a complete triplet of 3s, but 11 stands alone without forming a triplet. Therefore, 297 is not a perfect cube.
Now consider 2744. Its prime factorization is 2 × 2 × 2 × 7 × 7 × 7. We can group this as (2 × 7) repeated three times, which is 14 × 14 × 14, or 14³. Since we have complete triplets, 2744 is a perfect cube.
Sometimes we need to make a number into a perfect cube by multiplying or dividing it by another number. Let us see how this works.
Consider 3087. Its prime factors are 3 × 3 × 7 × 7 × 7. We have a complete triplet of 7s, but only two 3s. To make a perfect cube, we need one more 3 to complete the triplet. Therefore, 3087 must be multiplied by 3. The product becomes 3 × 3 × 3 × 7 × 7 × 7, which equals 21 × 21 × 21, or 21³.
Now for division. Take 6750, which factors as 2 × 5 × 5 × 5 × 3 × 3 × 3. Here we have complete triplets of 5s and 3s, but an extra 2 that does not form a triplet. So, 6750 must be divided by 2. The quotient becomes 5 × 5 × 5 × 3 × 3 × 3, which equals 15 × 15 × 15, or 15³.
Here are two useful properties to remember. Cubes of odd natural numbers are always odd: 1³ = 1, 3³ = 27, 5³ = 125, and so on. Cubes of even natural numbers are always even: 2³ = 8, 4³ = 64, 6³ = 216, and so on.
Now we move to cube roots. The cube root of a number is the value that, when cubed, gives you back the original number. In symbols, if ∛x = y, then y³ = x. Conversely, if y³ = x, then ∛x = y.
For example, since the cube of 3 is 27, the cube root of 27 is 3. Since 7³ = 343, we have ∛343 = 7. Similarly, 5³ = 125 implies ∛125 = 5.
The most reliable way to find cube roots uses prime factorization. Here is the three-step process. First, break the number down into its prime factors. Second, arrange identical primes into groups of three. Third, pick one prime from each group and multiply these together.
Let us find the cube root of 729. Factorizing 729, we get 3 × 3 × 3 × 3 × 3 × 3. We can group these as two complete triplets of 3. Taking one 3 from each triplet, we get 3 × 3, which equals 9. Therefore, ∛729 = 9.
Another example: find the cube root of 1728. The prime factorization gives us 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3. We have three complete triplets. Taking one from each: 2 × 2 × 3 equals 12. So, ∛1728 = 12.
Now let us explore cube roots of negative numbers. Take any positive integer m. When you cube its negative, you get –m³. This means ∛(–m³) = –m.
This leads to a key rule. The cube root of any negative perfect cube is negative. For instance, ∛(−8) = −2, since ∛8 = 2. Similarly, ∛(−1000) = −10. And ∛(−1) = −1.
Remember: cube root of a negative number is negative, and cube root of a positive number is positive.
Next, let us look at cube roots of products. The cube root of a product equals the product of the individual cube roots. That is, ∛(xy) = ∛x × ∛y. Similarly, for three numbers: ∛(xyz) = ∛x × ∛y × ∛z.
For example, ∛(8 × 125) equals ∛8 × ∛125, which is 2 × 5, giving 10. Another example: ∛(500 × 54) can be rewritten as ∛(1000 × 27). This equals ∛1000 × ∛27, giving 10 × 3, which is 30.
For negative products: ∛(–16 × 32) becomes ∛(–8 × 64), which equals –2 × 4, giving –8.
Now consider cube roots of fractions. For fractions, the cube root of the numerator divided by the cube root of the denominator gives the answer. That is, ∛(x/y) = ∛x/∛y.
For example, ∛(125/216) equals ∛125 / ∛216. This gives 5/6. Similarly, ∛(–8/27) equals –2/3. And ∛(64/–125) equals 4/(–5), which is –4/5.
Finally, let us tackle cube roots of decimal numbers. The approach is simple: rewrite the decimal as a fraction, then apply what you have learned about cube roots of fractions.
For example, to find ∛0.027, we write it as 27/1000. ∛27 is 3, and ∛1000 is 10. So we get 3/10, which is 0.3.
Similarly, ∛0.008 equals ∛(8/1000), which is 2/10, or 0.2. And ∛0.125 equals ∛(125/1000), giving 5/10, which is 0.5.
Let us now recap the key takeaways from this lesson.
First, the cube of a number m is m³, which equals m × m × m. Cubes of positive numbers are positive; cubes of negative numbers are negative.
Second, a perfect cube has every prime factor appearing in triplets of equal factors. To convert a number into a perfect cube, multiply or divide by the smallest number that completes all prime triplets.
Third, the cube root of x is the number whose cube is x, written as ∛x.
Fourth, to find cube roots by factorization: resolve into primes, group into triplets, and multiply one factor from each triplet.
Fifth, cube roots preserve signs: negative inputs give negative outputs. Also, cube root of a product equals the product of cube roots, and cube root of a fraction equals the quotient of cube roots.
Sixth, for decimal numbers, convert to fractions first, then apply the cube root.
That brings us to the end of our lesson on Cubes and Cube Roots. I hope you now feel confident in identifying perfect cubes, finding cube roots by prime factorization, and working with negative numbers, fractions, and decimals. Keep practicing these methods, and you will master them in no time. Thank you for listening, and see you in the next lesson.