SOLUTION: Use property 1 to simplify each of the radical expressions. Assume that all variables represent positive real numbers. 1. square root of 50 2. square root of 72x cubed

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Question 118059: Use property 1 to simplify each of the radical expressions. Assume that all variables represent positive real numbers.
1. square root of 50
2. square root of 72x cubed
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
#1

sqrt%2850%29 Start with the given expression

The goal of simplifying expressions with square roots is to factor the radicand into a product of two numbers. One of these two numbers must be a perfect square. When you take the square root of this perfect square, you will get a rational number.
So let's list the factors of 50
Factors:
1, 2, 5, 10, 25, 50


Notice how 25 is the largest perfect square, so lets factor 50 into 25*2


sqrt%2825%2A2%29 Factor 50 into 25*2

sqrt%2825%29%2Asqrt%282%29 Break up the square roots using the identity sqrt%28x%2Ay%29=sqrt%28x%29%2Asqrt%28y%29

5%2Asqrt%282%29 Take the square root of the perfect square 25 to get 5

So the expression sqrt%2850%29 simplifies to 5%2Asqrt%282%29

----------------------------
Check:
Notice if we evaluate the square root of 50 with a calculator we get

sqrt%2850%29=7.07106781186548

and if we evaluate 5%2Asqrt%282%29 we get

5%2Asqrt%282%29=7.07106781186548

This shows that sqrt%2850%29=5%2Asqrt%282%29. So this verifies our answer





#2


sqrt%2872%2Ax%5E3%29 Start with the given expression
sqrt%2836%2A2%2Ax%5E3%29 Factor 72 into 36%2A2

sqrt%2836%2A2%2Ax%2Ax%5E2%29 Factor x%5E3 into x%2Ax%5E2

sqrt%2836%29%2Asqrt%282%29%2Asqrt%28x%29%2Asqrt%28x%5E2%29 Break up the square root using the identity sqrt%28x%2Ay%29=sqrt%28x%29%2Asqrt%28y%29

6%2Asqrt%282%29%2Asqrt%28x%29%2Asqrt%28x%5E2%29 Take the square root of the perfect square 36 to get 6

6%2Asqrt%282%29%2Asqrt%28x%29%2Ax Take the square root of the perfect squares x%5E2 to get x

6%2Ax%2Asqrt%28x%29%2Asqrt%282%29 Rearrange the terms
6%2Ax%2Asqrt%282x%29 Group the square root terms


So sqrt%2872%2Ax%5E3%29 simplifies to 6%2Ax%2Asqrt%282x%29