Factor to find identical pairs.

[latex]\sqrt{3\cdot 3\cdot {{x}^{3}}\cdot {{x}^{3}}}[/latex]

Rewrite the pairs as perfect squares, note how we use the power rule for exponents to simplify [latex]x^6[/latex] into a square: [latex]{x^3}^2[/latex]

[latex]\sqrt{{{3}^{2}}\cdot {{\left( {{x}^{3}} \right)}^{2}}}[/latex]

Separate into individual radicals.

[latex]\sqrt{{{3}^{2}}}\cdot \sqrt{{{\left( {{x}^{3}} \right)}^{2}}}[/latex]

Simplify, using the rule that [latex]\sqrt{{{x}^{2}}}=\left|x\right|[/latex].

[latex]3\left|{{x}^{3}}\right|[/latex]

Answer

[latex]\sqrt{9{{x}^{6}}}=3\left|{{x}^{3}}\right|[/latex]

Separate factors; look for squared numbers and variables. Factor 100 into [latex]10\cdot10[/latex].

[latex]\sqrt{10\cdot 10\cdot {{x}^{2}}\cdot {{y}^{4}}}[/latex]

Factor [latex]y^{4}[/latex] into [latex]\left(y^{2}\right)^{2}[/latex].

[latex]\sqrt{10\cdot 10\cdot {{x}^{2}}\cdot {{({{y}^{2}})}^{2}}}[/latex]

Separate the squared factors into individual radicals.

[latex]\sqrt{{{10}^{2}}}\cdot \sqrt{{{x}^{2}}}\cdot \sqrt{{{({{y}^{2}})}^{2}}}[/latex]

Take the square root of each radical . Since you do not know whether x is positive or negative, use [latex]\left|x\right|[/latex] to account for both possibilities, thereby guaranteeing that your answer will be positive.

[latex]10\cdot\left|x\right|\cdot{y}^{2}[/latex]

Simplify and multiply.

[latex]10\left|x\right|y^{2}[/latex]

Answer

[latex]\sqrt{100{{x}^{2}}{{y}^{4}}}=10\left| x \right|{{y}^{2}}[/latex]

Look for squared numbers and variables. Factor 49 into [latex]7\cdot7[/latex], [latex]x^{10}[/latex] into [latex]x^{5}\cdot{x}^{5}[/latex], and [latex]y^{8}[/latex] into [latex]y^{4}\cdot{y}^{4}[/latex].

[latex]\sqrt{7\cdot 7\cdot {{x}^{5}}\cdot{{x}^{5}}\cdot{{y}^{4}}\cdot{{y}^{4}}}[/latex]

Rewrite the pairs as squares.

[latex]\sqrt{{{7}^{2}}\cdot{{({{x}^{5}})}^{2}}\cdot{{({{y}^{4}})}^{2}}}[/latex]

Separate the squared factors into individual radicals.

[latex]\sqrt{7^2}\cdot\sqrt{({x^5})^2}\cdot\sqrt{({y^4})^2}[/latex]

Take the square root of each radical using the rule that [latex]\sqrt{{{x}^{2}}}=\left|x\right|[/latex].

[latex]7\cdot\left|{{x}^{5}}\right|\cdot{{y}^{4}}[/latex]

Multiply.

[latex]7\left|{{x}^{5}}\right|{{y}^{4}}[/latex]

Answer

[latex]\sqrt{49{{x}^{10}}{{y}^{8}}}=7\left|{{x}^{5}}\right|{{y}^{4}}[/latex]

Factor to find variables with even exponents.

[latex]\sqrt{{{a}^{2}}\cdot a\cdot {{b}^{4}}\cdot{b}\cdot{{c}^{2}}}[/latex]

Rewrite [latex]b^{4}[/latex] as [latex]\left(b^{2}\right)^{2}[/latex].

[latex]\sqrt{{{a}^{2}}\cdot a\cdot {{({{b}^{2}})}^{2}}\cdot{ b}\cdot{{c}^{2}}}[/latex]

Separate the squared factors into individual radicals.

[latex]\sqrt{{{a}^{2}}}\cdot\sqrt{{{({{b}^{2}})}^{2}}}\cdot\sqrt{{{c}^{2}}}\cdot \sqrt{a\cdot b}[/latex]

Take the square root of each radical. Remember that [latex]\sqrt{{{a}^{2}}}=\left| a \right|[/latex].

[latex]\left| a \right|\cdot {{b}^{2}}\cdot\left|{c}\right|\cdot\sqrt{a\cdot b}[/latex]

Simplify and multiply. The entire quantity [latex]a{{b}^{2}}c[/latex] can be enclosed in the absolute value sign because [latex]b^2[/latex] will be positive anyway.

[latex]\left| a{{b}^{2}}c \right|\sqrt{ab}[/latex]

Answer

[latex]\sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}=\left| a{{b}^{2}}c\right|\sqrt{ab}[/latex]