To simplify the expression √35×√210:
Step 1: Prime Factorization
First, we need to find the prime factorization of both 35 and 210.
Prime factorization of 35:
35 = 5 × 7
Prime factorization of 210:
210 = 2 × 3 × 5 × 7
Step 2: Simplify the Radicands
Now, we simplify the radicands by extracting perfect square factors. We group them in pairs using the property √(a × b) = √a × √b.
√35 = √(5 × 7) = (√5) × (√7)
√210 = √(2 × 3 × 5 × 7) = (√2) × (√3) × (√5) × (√7)
Step 3: Combine Like Terms
We can now combine like terms under each square root symbol.
√35 × √210 = (√5) × (√7) × (√2) × (√3) × (√5) × (√7)
Step 4: Simplify
√35 × √210 = (√5 × √5) × (√2 × √3) × (√7 × √7)
Since √5 × √5 simplifies to 5, and √7 × √7 simplifies to 7, we have:
√35 × √210 = 5 × (√2 × √3) × 7
Now, we simplify √2 × √3:
√2 × √3 = √(2 × 3) = √6
Therefore:
√35 × √210 = 5 × √6 × 7
Final Answer: 35√6
Show all of your steps for how to simplify 35√×210−−√
. Your final answer should be in simplified radical form (not a decimal).
1 answer