Show all of your steps for how to simplify 35√×210−−√
. Your final answer should be in simplified radical form (not a decimal).
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
To simplify the expression 35√(210) - √(35), follow these steps:
Step 1: Factorize the numbers inside the radicals (square roots).
The prime factorization of 210 is 2 × 3 × 5 × 7.
The prime factorization of 35 is 5 × 7.
Step 2: Simplify the square roots individually.
√(210) can be simplified as √(2 × 3 × 5 × 7).
Since there are two 5's and one 7, we can take them out from under the square root.
√(210) = √(2 × 3 × 5 × 7) = √(2 × 3) × 25 √(7) = 5√(6) × √(7) = 5√(6 × 7) = 5√(42).
Similarly, √(35) can be simplified as √(5 × 7) = √(5) × √(7) = √(5 × 7) = √(35).
Step 3: Substitute the simplified values back into the original expression.
35√(210) - √(35) becomes 35(5√(42)) - √(35).
Step 4: Simplify further using basic operations.
35(5√(42)) = 175√(42).
Therefore, the simplified form of 35√(210) - √(35) is 175√(42).
To simplify the expression 35√(210) - √(35), we can follow these steps:
Step 1: Simplify the radicand (the number inside the square root) of each term.
√(210) can be simplifieed by factoring out the perfect square numbers:
√(210) = √(2 * 3 * 5 * 7)
Step 2: Simplify further by extracting the perfect square numbers:
√(210) = √(2 * 3 * 5 * 7) = √[(2 * 3) * (5 * 7)] = √(2^2 * 3^2 * 5 * 7)
= (2 * 3)√(5 * 7) = 6√(35)
Therefore, we have simplified the first term to 6√(35).
Step 3: Simplify the second term, √(35), using the same process as above:
√(35) = √(5 * 7) = √(5 * 7) = √(5 * 7) = 1√(5 * 7) = √(5 * 7) = √(35)
Therefore, the second term remains as √(35).
Step 4: Substitute the simplified terms back into the original expression:
35√(210) - √(35) = 35 * 6√(35) - √(35) = 210√(35) - √(35)
Step 5: Combine the terms with the same radical:
210√(35) - √(35) = (210 - 1)√(35) = 209√(35)
In simplified radical form, the final answer is 209√(35).