Final answer:
Simplify the expression 6 sqrt(15y⁴) * 2 sqrt(20y²) by first multiplying the coefficients and then the square roots. Simplify inside the square root, factor out perfect squares, and the simplified result is 120y³√(3).
Step-by-step explanation:
To simplify the expression 6 sqrt(15y⁴) * 2 sqrt(20y²), we apply the properties of square roots and exponents.
First, we'll rewrite the square roots as powers: sqrt(15y⁴) = (15y⁴)¹⁄₂ and sqrt(20y²) = (20y²)¹⁄₂. Keeping in mind that √(x) = x¹⁄₂, we can use the property of exponents which states that (xᵗʰ)ᵗᵉⁿ = xᵗⁿ×ᵉⁿ. We simplify the expression as follows:
- Multiply the numerical coefficients: 6 * 2 = 12.
- Multiply the square roots: √(15y⁴) * √(20y²) = √(15*20*y⁴*y²).
- Simplify inside the square root: √(15*20*y⁴*y²) = √(300y⁶).
- Now we look for perfect squares inside the square root. Since 100 is a perfect square and it's a factor of 300 (300 = 100 * 3), and y⁶ is also a perfect square (since 6 is even), we can rewrite √(300y⁶) as √(100*3*y´*y²) = √(100)*√(y´)*√(3)*√(y²).
- We know that √(100) = 10 and √(y´) = y², and √(y²) = y, so we can rewrite the original expression as 12 * 10 * y² * √(3) * y = 120y³√(3).
Therefore, the simplified expression is 120y³√(3).