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Unit 6.4 Simplifying Radical Expressions with Variables

Simplifying with Variables

Simplifying radical expressions with variables follows the same principles as those with just numbers, but requires careful attention to the variables and their exponents.

Simplification Steps:

  1. Factor the radicand (the expression inside the radical) into prime factors, including the variables using the product or quotient rules of radical expressions
  2. Separate the factors into perfect squares (or cubes, etc., depending on the root) and the remaining factors.
  3. Simplify the radical by taking the root of the perfect squares (or cubes, etc.).

Example 1: Simplify a radical expression

Simplify [latex]\sqrt{50x^4}[/latex]

Factor the radicand

[latex]\ 50x^4 = 25\times2\times x^4[/latex]

Recognize the perfect roots

[latex]\ 25\ and \ x^4[/latex]

Simplify

[latex]\sqrt{50x^4} = \sqrt{25 \times 2 \times x^4} = \sqrt{25} \times \sqrt{2} \times \sqrt{x^4} = 5 \times \sqrt{2} \times x^2 = 5x^2\sqrt{2}[/latex]

Example 2: Simplify a radical expression

Simplify [latex]\sqrt[3]{8x^6y^3}[/latex]

Factor the radicand

[latex]\ 8x^6y^3 = 2^3 \times x^6 \times y^3[/latex]

Recognize the perfect roots

[latex]\ 2^3\, x^6, and \ y^3[/latex]

Simplify

[latex]\sqrt[3]{8x^6y^3} = \sqrt[3]{2^3 \times x^6 \times y^3} = \sqrt[3]{2^3} \times \sqrt[3]{x^6} \times \sqrt[3]{y^3} = 2 \times x^2 \times y = 2x^2y[/latex]

Example 3: Simplifying a radical expression

Simplify [latex]\sqrt{4x^5}[/latex]

Write in fractional exponent form

[latex]\sqrt{4x^5}\ =\ {(4x^5)}^\frac{1}{2}[/latex]

Distribute the 1/2 power

[latex]4^\frac{1}{2}\ \cdot x^\frac{5}{2}[/latex]

Since 5 is not evenly divisible by 2, break apart the x fraction so that we get at least 1 value divisible by 2

[latex](4^\frac{1}{2}\ \cdot\ x^\frac{4}{2}\ \cdot\ x^\frac{1}{2})[/latex]

Simplify

[latex]2\ \cdot\ x^2\cdot x^\frac{1}{2}=2x^2\sqrt{x}[/latex]

 

 

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