Answer:
Explanation:
Simplifying square roots
Example
Let's simplify \sqrt{75}
75
square root of, 75, end square root by removing all perfect squares from inside the square root.
We start by factoring 757575, looking for a perfect square:
75=5\times5\times3=\blueD{5^2}\times375=5×5×3=5
2
×375, equals, 5, times, 5, times, 3, equals, start color #11accd, 5, squared, end color #11accd, times, 3.
We found one! This allows us to simplify the radical:
\begin{aligned} \sqrt{75}&=\sqrt{\blueD{5^2}\cdot3} \\\\ &=\sqrt{\blueD{5^2}} \cdot \sqrt{{3}} \\\\ &=5\cdot \sqrt{3} \end{aligned}
75
=
5
2
⋅3
=
5
2
⋅
3
=5⋅
3
So \sqrt{75}=5\sqrt{3}
75
=5
3
square root of, 75, end square root, equals, 5, square root of, 3, end square root.
Want another example like this? Check out this video.
Practice
PROBLEM 1.1
Simplify.
Remove all perfect squares from inside the square root.
{\sqrt[]{12}}=
12
=root, start index, end index, equals
Explain
Want to try more problems like these? Check out this exercise.
Simplifying square roots with variables
Example
Let's simplify \sqrt{54x^7}
54x
7
square root of, 54, x, start superscript, 7, end superscript, end square root by removing all perfect squares from inside the square root.
First, we factor 545454:
54=3\cdot 3\cdot 3\cdot 2=3^2\cdot 654=3⋅3⋅3⋅2=3
2
⋅654, equals, 3, dot, 3, dot, 3, dot, 2, equals, 3, squared, dot, 6
Then, we find the greatest perfect square in x^7x
7
x, start superscript, 7, end superscript:
x^7=\left(x^3\right)^2\cdot xx
7
=(x
3
)
2
⋅xx, start superscript, 7, end superscript, equals, left parenthesis, x, cubed, right parenthesis, squared, dot, x
And now we can simplify:
\begin{aligned} \sqrt{54x^7}&=\sqrt{3^2\cdot 6\cdot\left(x^3\right)^2\cdot x} \\\\ &=\sqrt{3^2}\cdot \sqrt6 \cdot\sqrt{\left(x^3\right)^2}\cdot \sqrt x \\\\ &=3\cdot\sqrt6\cdot x^3\cdot\sqrt x \\\\ &=3x^3\sqrt{6x} \end{aligned}
54x
7
=
3
2
⋅6⋅(x
3
)
2
⋅x
=
3
2
⋅
6
⋅
(x
3
)
2
⋅
x
=3⋅
6
⋅x
3
⋅
x
=3x
3
6x
Practice
PROBLEM 2.1
Simplify.
Remove all perfect squares from inside the square root.
\sqrt{20x^8}=
20x
8
=square root of, 20, x, start superscript, 8, end superscript, end square root, equals
Explain
Want to try more problems like these? Check out this exercise.
More challenging square root expressions
PROBLEM 3.1
Simplify.
Combine like terms and remove all perfect squares from inside the square roots.
2\sqrt{7x}\cdot 3\sqrt{14x^2}=2
7x
⋅3
14x
2
=2, square root of, 7, x, end square root, dot, 3, square root of, 14, x, squared, end square root, equals
Explain
Want to try more problems like these? Check out this exercise.