Cube root through prime factorisation method

Today, we are going to explore how we can find the cube roots of numbers using prime factorisation. Can anyone tell me what a cube root is?

Isn't it the number that when multiplied by itself three times gives the original number?

Exactly, well done! For example, $3^3 = 27$, so the cube root of 27 is 3. We represent this as $\sqrt[3]{27} = 3$.

What if we have a bigger number, like 64?

Good question! We will show you how to find the cube root of larger numbers by breaking them down into their prime factors.

Could you give us an example?

Sure! Let's take 3375. I’ll show you how to factor it and find its cube root. So we find the prime factorisation first.

The factorisation is $3375 = 3 \times 3 \times 3 \times 5 \times 5 \times 5 = 3^3 \times 5^3 = (3 \times 5)^3$. So what's the cube root of 3375?

That would be 15, right?

Correct! You're all doing great.

Now that we have our first example, let’s try another number! What do you think is the cube root of 74088?

How do we start?

Let's begin by performing the prime factorisation of 74088. After the factorisation, we have $74088 = 2^3 \times 3^3 \times 7^3$. Can someone tell me how to find the cube root from here?

Combine the factors from groups of three, right?

Yes! By combining those factors, we can rewrite the expression as $(2 \times 3 \times 7)^3$. Now what do we get?

The cube root of 74088 would be 42!

Absolutely correct! Remember the formula that links these concepts: $\sqrt[3]{a^3} = a$ helps summarize what we just did.

Let’s practice together. How about we find the cube root of 8000? What’s the first step?

We should factor 8000 into its prime factors!

That’s right! The factorization is $8000 = 2^6 \times 5^3$. Now, can you tell me how to find the cube root?

We take the cube root of each factor? So $\sqrt[3]{2^6} = 2^2$ and $\sqrt[3]{5^3} = 5$.

Perfect! So what is the cube root of 8000?

That’s 20!

Excellent work, everyone! Have you noticed how the cube roots relate to their prime bases?

Yes, understanding it like that makes it easier!