simplify each radical expression: 1. sqrt 75 + sqrt3 2. sqrt 7 (sqrt14 + sqrt 3) please walk me through this.
1. sqrt75 + sqrt3
sqrt(25*3) + sqrt3
5*sqrt3 + sqrt3. = 6*sqrt3.
2. sqrt7(sqrt14+sqrt3)
sqrt7(sqrt(7*2)+sqrt3)
sqrt7(sqrt7*sqrt2+sqrt3)
7*sqrt2 + sqrt7*sqrt3
7*sqrt2 + sqrt21.
To simplify radical expressions, we want to break down the numbers inside the square roots into their factors, and then simplify as much as possible.
1. √75 + √3:
First, let's break down 75 into its factors: 75 = 25 * 3.
Now, rewrite the expression using the factors: √(25 * 3) + √3.
Since the square root of 25 is 5, the expression becomes: 5√3 + √3.
Combining like terms, we have: (5 + 1)√3 = 6√3.
So, the simplified form of √75 + √3 is 6√3.
2. √7 (√14 + √3):
Similar to the previous example, let's break down both 14 and 3 into their factors.
14 = 2 * 7, and 3 is a prime number, so there are no factors to simplify.
Now, rewrite the expression using the factors: √7 (√(2 * 7) + √3).
Since the square root of 7 cannot be simplified further, we leave it as it is.
The square root of 2 multiplied by the square root of 7 is √(2 * 7) = √14.
Therefore, the expression becomes: √7 (√14 + √3).
Finally, combining like terms, we have: √7 √14 + √7 √3.
Multiplying these square roots, we get: √(7 * 14) + √(7 * 3).
Therefore, the simplified form is: √98 + √21.
To summarize:
1. √75 + √3 simplifies to 6√3.
2. √7 (√14 + √3) simplifies to √98 + √21.