Using prime factorisation method, find the cube root of 1024 x 8788.

Using prime factorisation method, find the cube root of 1024 x 8788.

The Correct Answer and Explanation is:

✅ Answer:

Using prime factorization, the cube root of 1024×87881024 \times 87881024×8788 is:1024×87883=213×22×1093=215×1093\sqrt[3]{1024 \times 8788} = \sqrt[3]{2^{13} \times 2^2 \times 109} = \sqrt[3]{2^{15} \times 109}31024×8788​=3213×22×109​=3215×109​

Now simplify:215×1093=25×1093=32×1093\sqrt[3]{2^{15} \times 109} = 2^5 \times \sqrt[3]{109} = 32 \times \sqrt[3]{109}3215×109​=25×3109​=32×3109​

🟩 Final Answer: 32×1093\boxed{32 \times \sqrt[3]{109}}32×3109​​ (This is the simplest exact form)

---

📘 Detailed Explanation

To find the cube root of 1024×87881024 \times 87881024×8788 using prime factorization, we first factorize each number into its prime components.

Step 1: Prime factorization of 1024

1024=2×2×2×…(10 times)=2101024 = 2 \times 2 \times 2 \times \dots \text{(10 times)} = 2^{10}1024=2×2×2×…(10 times)=210

Step 2: Prime factorization of 8788

Let’s divide 8788 by 2 (smallest prime number):8788÷2=43944394÷2=21978788 \div 2 = 4394 \\ 4394 \div 2 = 21978788÷2=43944394÷2=2197

Now check 2197. It’s not divisible by 2, 3, or 7. Try 13:13×169=219713 \times 169 = 219713×169=2197

Then:13×13=169⇒13×13×13=2197⇒2197=13313 \times 13 = 169 \Rightarrow 13 \times 13 \times 13 = 2197 \Rightarrow 2197 = 13^313×13=169⇒13×13×13=2197⇒2197=133

So,8788=22×1338788 = 2^2 \times 13^38788=22×133

Step 3: Multiply the two numbers:

1024×8788=210×(22×133)=212×1331024 \times 8788 = 2^{10} \times (2^2 \times 13^3) = 2^{12} \times 13^31024×8788=210×(22×133)=212×133

Now take the cube root:212×1333=(24)3×1333=24×13=16×13=208\sqrt[3]{2^{12} \times 13^3} = \sqrt[3]{(2^4)^3 \times 13^3} = 2^4 \times 13 = 16 \times 13 = \boxed{208}3212×133​=3(24)3×133​=24×13=16×13=208​

🔁 Correction to earlier steps: I mistakenly used 109 before — the actual prime factor of 8788 is 13, not 109.

---

✅ Correct Final Answer: 208\boxed{208}208​

This method shows how breaking down large numbers into their prime factors allows for easy root extraction by grouping exponents into multiples of 3.