Algebra

5th_root(486x^10y^5)

Step-by-step solution with explanation

Final Answer

Step-by-step solution

1

Rewrite the expression under the radical

We want to simplify the 5th root of 486x¹⁰y⁵. We'll break the expression into parts we can take the 5th root of easily.
2

Factor 486 into prime factors

We divide 486 by small primes. 486 ÷ 2 = 243, and 243 = 3⁵. So 486 = 2 · 3⁵. This lets us pull out the perfect 5th power factor of 3⁵.
3

Rewrite the full expression using factors

We replace 486 with 2 · 3⁵ and keep x¹⁰ and y⁵ as they are. Now it's easy to see which parts are perfect 5th powers.
4

Identify all perfect 5th power factors

3⁵ is a perfect 5th power. x¹⁰ = (x²)⁵ is a perfect 5th power. y⁵ is a perfect 5th power. The leftover factor is just 2.
5

Apply the 5th root to each perfect power

We split the 5th root across each factor using the product rule for radicals. Each perfect 5th power simplifies cleanly.
6

Simplify each 5th root

The 5th root of 3⁵ is 3. The 5th root of (x²)⁵ is x². The 5th root of y⁵ is y. The 5th root of 2 stays as is since 2 is not a perfect 5th power.
7

Write the final simplified expression

We multiply the simplified pieces together. The result is 3x²y times the 5th root of 2. This is fully simplified because no perfect 5th power factors remain under the radical.

Understanding this problem

Learning Insight

The 5th root is the inverse of raising to the 5th power. When you split an exponent like x¹⁰ as (x²)⁵, you're recognizing it as a perfect 5th power so it comes out of the radical cleanly. The key idea is always factoring out the largest perfect nth power from under an nth root.

Quick Tip

For any variable xᵃ under a 5th root, divide the exponent by 5. The quotient is the outside exponent, and the remainder stays inside. Here, x¹⁰ ÷ 5 = 2 remainder 0, so x² comes out and nothing stays in.

Common Mistake

Students often forget to fully factor the number under the radical. For example, they might leave ⁵√486 as is instead of breaking it into 2 · 3⁵ and pulling out the 3. Always check if the coefficient contains a perfect 5th power factor.