Final answer:
To simplify √42a⁴b⁶, you identify the perfect square factors a² and b³ within the radicand. After simplifying, the expression becomes a²b³√(42), as 42 has no perfect square factors other than 1.
Step-by-step explanation:
To simplify the square root of 42a⁴b⁶, we need to identify the perfect square factors within the radicand and remove them from inside the square root.
First, we factor the number 42 into its prime factors: 2 × 3 × 7.
Then, we consider the variables a⁴ and b⁶. Since a⁴ is equivalent to (a²)² and b⁶ is equivalent to (b³)², we recognize these as perfect squares. Therefore, we can simplify the expression:
√(42a⁴b⁶)
= √(2 × 3 × 7 × (a²)² × (b³)²)
= √(2) × √(3) × √(7) × a² × b³
= a²b³√(42)
We cannot simplify the √(42) any further since 42 does not have perfect square factors apart from 1, which does not affect the value.
Therefore, the simplified expression, with all perfect squares removed from inside the square root, is a²b³√(42).
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