SOLUTION: express 5√72 in simplest radical form √ = radical sign find the sum of: √120 + √75

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#1

Start with the given expression


Let's simplify

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 72
Factors:
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72


Notice how 36 is the largest perfect square, so lets factor 72 into 36*2


Factor 72 into 36*2

Break up the square roots using the identity

Take the square root of the perfect square 36 to get 6

So the expression simplifies to



Now reintroduce the 5 to get . Now multiply to get




So simplifies to

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Check:
Notice if we evaluate with a calculator we get



and if we evaluate we get



This shows that . So this verifies our answer

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#2
Start with the given expression


Factor 120 into 15*8 and factor 75 into 15*5


Break up the square roots



Factor out the common term of


So simplifies to