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### Course: Algebra 1>Unit 11

Lesson 3: Simplifying square roots

# Simplifying square-root expressions

Worked examples of taking expressions with square roots and taking all of the perfect squares out of the square roots. For example, 2√(7x)⋅3√(14x²) can be written as 42x√(2x).

## Want to join the conversation?

• Solve this, please square root of (x+ 15) + square root of (x) = 15
• First, you are in the wrong section of lessons. You have a radical equation, not a radical expression.

For a problem more like yours, I would suggest you look at the 2nd problem at this link: http://www.purplemath.com/modules/solverad3.htm

I'll get you started on your equation: √(x+15) + √(x) = 15
1) I would move one radical to the other side. I think it is less confusing. The link above keeps them both on the same side.
Subtract √(x): √(x+15) = 15 - √(x)
2) Square both sides: [√(x+15)]^2 = [15 - √(x) ]^2
3) Simplify left side. FOIL or use extended distribution on the right side to eliminate the exponents
x + 15 = 225 - 30√(x) + x
4) Subtract x: 15 = 225 - 30√(x)
5) Subtract 225: -210 = - 30√(x)
6) Divide by -30: 7 = √(x)
7) Square both sides again: 7^2 = √(x)^2
8) Simplify: 49 = x
9) Check answer back in original equation to verify that it isn't an extraneous solution.

hope this helps.
• At , you state the answer: 6xz√2xz.
However, what happened to the rule that x or z can't be negative?
Did we assume all variables were greater than or equal to zero in the beginning?
I know we don't need absolute value of the variable if it is x^2, or x^4. But in this case, it was just 6xz√2xz. Both x and z were singular, and if one of them were negative(and the other positive), wouldn't this answer be incorrect without a restraint?
• When we write the normal square root sign,we assume this expression to be a PRINCIPLE ROOT, in which all variables are positive (it is positive itself), since it is basically impossible to square any number to reach a negative value. However, it is possible in cubes.
In the answer 6xz√2xz, we are evidently using a principle root. This means that either both variables are positive or both are negative, since the resolution of one variable being negative cannot satify the principle root. So, if both variables are both negative, or both positive, the result would stay the same.
• I don't get this!
• at , how do you solve the equation radical 75yz to the second power? i watched the video but i am still a little confused.
• What is to the second power, √(75yz^2)? I think this is what you mean, so 75 breaks down to 25z^2 (perfect square) and 3y (non-perfect square), √(25z^2)*√(3y) = 5z√3y.
The other choice is √(75yz)^2 in which case the square and square root cancel to give 75yz.
• Hello, I'm hoping that someone can show me the way!

I am working through the following square root simplification problem:

Square Root of 108a^6

108 can be prime factored into: 2*2*3*3*3

I then broke it down as:

Sq rt of 2^2
Sq rt of 3^3
Sq rt of a^6

I then came up with the following:

2*3*a^3
or further simplified: 6a^3

The Khan answer (which I of course presume is correct) came up with the following simplification:

6a^3 sq rt of 3

It looks like rather than combining like integers and adding exponents, Khan multiplied 2*2*3*3*3= 6^2*3

----
Is there a rule or a step that I'm missing here that brings me to an incorrect answer?

• Your approach is ok. But, the sqrt(3^3) will not equal 3. You need to look for perfect squares which would have an even exponent. Since this exponent is odd, regroup. Split the factors into sqrt(3^2) sqrt(3) = 3 sqrt(3)
2*3*a^3 sqrt(3)
Multiply factors outside to get:
6 a^3 sqrt(3)

Hope this helps.
• i do not understand any of it, just since like you just make up numbers, how is any of is simplified?
• I understood everything, it was easy to understand
• good for you bud
• In this video, Sal identified a variable as a perfect square if its exponent is even. But what if the variable itself is a decimal number? For example, let's say x=1.5
x^2=2.25.
And 2.25 is not a perfect square. So what's going on, can someone please explain?
• Actually, 2.25 IS a perfect square!
First, let's think of 2.25 as 225. The square root of that is 15. This means that the square root of 2.25 is 1.5.
Here is another example.
x^2 = 0.36
Let's (once again) think of it as a whole number, 36. The square root of 36 is 6, so the square root of 0.36 is 0.6.

Hope this helps you, Issac Newton!