Question

square roots of monomials
to find the square root of a number we can think of the number as the area of a the square root as the length of a side.
sqrt{676}=26
what would be meant by ( sqrt{x^{2}} )?
think of ( x^{2} ) as the area of a square.
the length of a side must be ( x ), so
sqrt{x^{2}}=x
we can find the square root of any expression that we can write as a square. we will assume that all of our variables stand for positive numbers.
find each square root by writing the expression as a square.
sqrt{x^{10}}=sqrt{x^{5}cdot x^{5}} = x^{5}
sqrt{x^{8}}=
sqrt{a^{12}}=
sqrt{c^{6}}=
sqrt{25n^{2}}=sqrt{(5n)^{2}} = 5n
sqrt{9y^{2}}=
sqrt{100x^{2}}=
sqrt{81a^{4}}=
sqrt{121y^{6}}=
sqrt{x^{2}y^{4}}=
sqrt{a^{6}b^{2}}=
sqrt{n^{4}m^{6}}=
12

Explanation:

Step1: Recall square - root property

For $\sqrt{x^{n}}$, if $n$ is even and $x>0$, $\sqrt{x^{n}}=x^{\frac{n}{2}}$. Also, $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ for $a,b\geq0$.

Step2: Solve $\sqrt{x^{8}}$

Since $n = 8$ (even), $\sqrt{x^{8}}=x^{\frac{8}{2}}=x^{4}$.

Step3: Solve $\sqrt{a^{12}}$

As $n = 12$ (even), $\sqrt{a^{12}}=a^{\frac{12}{2}}=a^{6}$.

Step4: Solve $\sqrt{c^{6}}$

With $n = 6$ (even), $\sqrt{c^{6}}=c^{\frac{6}{2}}=c^{3}$.

Step5: Solve $\sqrt{9y^{2}}$

We know $\sqrt{9y^{2}}=\sqrt{9}\cdot\sqrt{y^{2}}$. Since $\sqrt{9} = 3$ and $\sqrt{y^{2}}=y$ ($y>0$), $\sqrt{9y^{2}}=3y$.

Step6: Solve $\sqrt{100x^{2}}$

$\sqrt{100x^{2}}=\sqrt{100}\cdot\sqrt{x^{2}}$. As $\sqrt{100}=10$ and $\sqrt{x^{2}}=x$ ($x>0$), $\sqrt{100x^{2}}=10x$.

Step7: Solve $\sqrt{81a^{4}}$

$\sqrt{81a^{4}}=\sqrt{81}\cdot\sqrt{a^{4}}$. Since $\sqrt{81}=9$ and $\sqrt{a^{4}}=a^{2}$ ($a>0$), $\sqrt{81a^{4}}=9a^{2}$.

Step8: Solve $\sqrt{121y^{6}}$

$\sqrt{121y^{6}}=\sqrt{121}\cdot\sqrt{y^{6}}$. As $\sqrt{121}=11$ and $\sqrt{y^{6}}=y^{3}$ ($y>0$), $\sqrt{121y^{6}}=11y^{3}$.

Step9: Solve $\sqrt{x^{2}y^{4}}$

$\sqrt{x^{2}y^{4}}=\sqrt{x^{2}}\cdot\sqrt{y^{4}}$. Since $\sqrt{x^{2}}=x$ and $\sqrt{y^{4}}=y^{2}$ ($x,y>0$), $\sqrt{x^{2}y^{4}}=xy^{2}$.

Step10: Solve $\sqrt{a^{6}b^{2}}$

$\sqrt{a^{6}b^{2}}=\sqrt{a^{6}}\cdot\sqrt{b^{2}}$. As $\sqrt{a^{6}}=a^{3}$ and $\sqrt{b^{2}}=b$ ($a,b>0$), $\sqrt{a^{6}b^{2}}=a^{3}b$.

Step11: Solve $\sqrt{n^{4}m^{6}}$

$\sqrt{n^{4}m^{6}}=\sqrt{n^{4}}\cdot\sqrt{m^{6}}$. Since $\sqrt{n^{4}}=n^{2}$ and $\sqrt{m^{6}}=m^{3}$ ($n,m>0$), $\sqrt{n^{4}m^{6}}=n^{2}m^{3}$.

Answer:

1. $\sqrt{x^{8}}=x^{4}$
2. $\sqrt{a^{12}} = a^{6}$
3. $\sqrt{c^{6}}=c^{3}$
4. $\sqrt{9y^{2}} = 3y$
5. $\sqrt{100x^{2}}=10x$
6. $\sqrt{81a^{4}} = 9a^{2}$
7. $\sqrt{121y^{6}}=11y^{3}$
8. $\sqrt{x^{2}y^{4}}=xy^{2}$
9. $\sqrt{a^{6}b^{2}}=a^{3}b$
10. $\sqrt{n^{4}m^{6}}=n^{2}m^{3}$