Simplify the radical expressions:
(1).sqrt75+sqrt3.
(2).sqrt7(sqrt14+sqrt3
1) factor 75 to be 25 times 3. You can then take the square root of 25 and you will be left with a sqrt of 3. Now, you have two like terms that you can combine.
Write 14 as 7 times 2.
Multiply the sqrt of 7 and but the 7 under the radical with the 7 times 2 and also under the radical sign with the 3.
Can you please show me how to write this because I don't follow what your saying
To simplify radical expressions, we need to look for perfect square factors within the radical and simplify them. Let's break down the given expressions step by step:
(1) √75 + √3:
First, let's find the perfect square factors of 75. The factors of 75 are 1, 3, 5, 15, 25, and 75. None of them are perfect squares.
However, we can simplify radical expressions by breaking down the number under the square root into its prime factors. The prime factorization of 75 is 5 × 5 × 3, which can be written as √(5^2 × 3).
Using the product rule of radicals, we can split the square root:
√(5^2 × 3) = √(5^2) × √3 = 5√3.
So, the simplified form of √75 + √3 is 5√3.
(2) √7 (√14 + √3):
Similarly, let's identify the perfect square factors within the expression.
For simplification, we can break down 14 and 3 into their prime factors:
The prime factorization of 14 is 2 × 7, and the prime factorization of 3 is 3.
Using the product rule of radicals, we can break down the expression:
√7 (√(2 × 7) + √3) = √7 (√2 × √7 + √3).
Next, simplify further:
√7 (√2 × √7 + √3) = √7 (√(2 × 7) + √3) = √7 (√14 + √3).
So, the simplified form of √7 (√14 + √3) is √7 (√14 + √3).