Final answer:
The expression (√3vw⁵) √(15v⁴w⁸) simplifies to 3v³w⁶√(5) by multiplying the terms inside the roots, combining like terms, and extracting perfect squares.
Step-by-step explanation:
To simplify the given expression √(3vw⁵) √(15v⁴w⁸), we first need to multiply the terms inside the square roots together. We can use the property of square roots that √(x)√(y) = √(x·y). Applying this, we get:
√(3vw⁵) √(15v⁴w⁸) = √(3·15·v·w⁵·v⁴·w⁸).
Now, we simplify inside the square root by multiplying the numbers and adding the exponents of like terms:
√(45v⁵v⁴w⁵w⁸) = √(45v⁹w¹³).
Next, we look for perfect squares inside the square root and take them outside the radical. We can rewrite 45 as 3·15, and notice that 3 is not a perfect square but 15 is 3·5, with 5 being a perfect square. So we can take out the v⁹ and w¹³ as v³w⁶ since (v³)² = v⁶ and (w⁶)² = w¹³2:
√(45v⁹w¹³) = √(3·5·v·v¸·w·w¹³2) = 3v³w⁶√(5).
Hence, the simplified form of the expression is 3v³w⁶√(5), which corresponds to option a) 3v³w⁷√(5).