Example

Simplify. [latex] a(3a-5)[/latex]

Answer: Use the Distributive Property of Multiplication over Subtraction.

[latex]\begin{array}{c}a(3a)-a(5)\\=3a^2-5a\end{array}[/latex]

Example

Simplify. [latex] \sqrt{x}(3\sqrt{x}-5)[/latex]

Answer: Use the Distributive Property of Multiplication over Subtraction.

[latex] \sqrt{x}(3\sqrt{x})-\sqrt{x}(5)[/latex]

Apply the rules of multiplying radicals: [latex] \sqrt{a}\cdot \sqrt{b}=\sqrt{ab}[/latex] to multiply [latex] \sqrt{x}(3\sqrt{x})[/latex].

[latex] 3\sqrt{{{x}^{2}}}-5\sqrt{x}[/latex]

Be sure to simplify radicals when you can: [latex] \sqrt{{{x}^{2}}}=\left| x \right|[/latex], so [latex] 3\sqrt{{{x}^{2}}}=3\left| x \right|[/latex]. The answer is [latex]3\left| x \right|-5\sqrt{x}[/latex]

Example

Simplify. [latex] 7\sqrt{x}\left( 2\sqrt{xy}+\sqrt{y} \right)[/latex]

Answer: Use the Distributive Property of Multiplication over Addition to multiply each term within parentheses by [latex] 7\sqrt{x}[/latex].

[latex] 7\sqrt{x}\left( 2\sqrt{xy} \right)+7\sqrt{x}\left( \sqrt{y} \right)[/latex]

Apply the rules of multiplying radicals.

[latex] 7\cdot 2\sqrt{{{x}^{2}}y}+7\sqrt{xy}[/latex]

[latex] \sqrt{{{x}^{2}}}=\left| x \right|[/latex], so [latex] \left| x \right|[/latex] can be pulled out of the radical.

[latex] 14|x|\sqrt{y}+7\sqrt{xy}[/latex]

Example

Simplify. [latex] \sqrt[3]{a}\left( 2\sqrt[3]{{{a}^{2}}}-4\sqrt[3]{{{a}^{5}}}+8\sqrt[3]{{{a}^{8}}} \right)[/latex]

Answer: Use the Distributive Property.

[latex] \sqrt[3]{a}\left( 2\sqrt[3]{{{a}^{2}}} \right)-\sqrt[3]{a}\left( 4\sqrt[3]{{{a}^{5}}} \right)+\sqrt[3]{a}\left( 8\sqrt[3]{{{a}^{8}}} \right)[/latex]

Apply the rules of multiplying radicals.

[latex] \begin{array}{c}2\sqrt[3]{a\cdot {{a}^{2}}}-4\sqrt[3]{a\cdot {{a}^{5}}}+8\sqrt[3]{a\cdot {{a}^{8}}}\\2\sqrt[3]{{{a}^{3}}}-4\sqrt[3]{{{a}^{6}}}+8\sqrt[3]{{{a}^{9}}}\end{array}[/latex]

Identify cubes in each of the radicals.

[latex] 2\sqrt[3]{{{a}^{3}}}-4\sqrt[3]{{{\left( {{a}^{2}} \right)}^{3}}}+8\sqrt[3]{{{\left( {{a}^{3}} \right)}^{3}}}[/latex]

The solution is [latex]2a-4{{a}^{2}}+8{{a}^{3}}[/latex].

Example

Multiply. [latex] \left( 2x+5 \right)\left( 3x-2 \right)[/latex]

Answer: Use the Distributive Property.

[latex]\begin{array}{l}\text{First}:\,\,\,\,\,\,\,\,\,\,\,2x\cdot 3x=6{{x}^{2}}\\\text{Outside}:\,\,\,2x\cdot \left( -2 \right)=-4x\\\text{Inside}:\,\,\,\,\,\,\,\,5\cdot 3x=15x\\\text{Last}:\,\,\,\,\,\,\,\,\,\,\,\,5\cdot \left( -2 \right)=-10\end{array}[/latex]

Record the terms, and then combine like terms.

[latex] 6{{x}^{2}}-4x+15x-10[/latex]

[latex]6{{x}^{2}}+11x-10[/latex]

Example

Multiply. [latex] \left( 2\sqrt{b}+5 \right)\left( 3\sqrt{b}-2 \right),\,\,b\ge 0[/latex]

Answer: Use the Distributive Property to multiply. Simplify using [latex] \sqrt{x}\cdot \sqrt{x}=x[/latex].

[latex]\begin{array}{l}\text{First}:\,\,\,\,\,\,\,\,\,\,\,2\sqrt{b}\cdot 3\sqrt{b}=2\cdot 3\cdot \sqrt{b}\cdot \sqrt{b}=6b\\\text{Outside}:\,\,\,2\sqrt{b}\cdot \left( -2 \right)=-4\sqrt{b}\\\text{Inside}:\,\,\,\,\,\,\,\,5\cdot 3\sqrt{b}=15\sqrt{b}\\\text{Last}:\,\,\,\,\,\,\,\,\,\,\,\,5\cdot \left( -2 \right)=-10\end{array}[/latex]

Record the terms, and then combine like terms.

[latex] 6b-4\sqrt{b}+15\sqrt{b}-10[/latex]

[latex]6b+11\sqrt{b}-10[/latex]

Example

Multiply. [latex] \left( 4{{x}^{2}}+\sqrt[3]{x} \right)\left( \sqrt[3]{{{x}^{2}}}+2 \right)[/latex]

Answer: Use FOIL to multiply.

[latex]\begin{array}{l}\text{First}:\,\,\,\,\,\,\,\,\,\,\,4x^{2}\cdot\sqrt[3]{x^{2}}=4x^{2}\sqrt[3]{x^{2}}\\\text{Outside}:\,\,\,4x^{2}\cdot 2=8x^{2}\\\text{Inside}:\,\,\,\,\,\,\,\,\sqrt[3]{x}\cdot\sqrt[3]{x^{2}}=\sqrt[3]{x^{2}\cdot x}=\sqrt[3]{x^{3}}=x\\\text{Last}:\,\,\,\,\,\,\,\,\,\,\,\,\sqrt[3]{x}\cdot 2=2\sqrt[3]{x}\end{array}[/latex]

Record the terms, and then combine like terms (if possible). Here, there are no like terms to combine.

[latex] 4{{x}^{2}}\sqrt[3]{{{x}^{2}}}+8{{x}^{2}}+x+2\sqrt[3]{x}[/latex]