About Rationalizing Denominators:
A simplified radical expression does not contain any radicals in the denominator. The process we use to clear a denominator of its radical is known as rationalizing the denominator. We rationalize the denominator by multiplying the numerator and denominator by a radical that will transform the denominator into a rational number. In some cases, we will face a two-term denominator that contains radicals. For this scenario, we can’t use the same methods from rationalizing with a single-term radical in the denominator. To rationalize a binomial denominator, we multiply numerator and denominator by the conjugate of the denominator.
Test Objectives
- Demonstrate an understanding of the rules for simplifying a radical
- Demonstrate the ability to rationalize a denominator with a square root
- Demonstrate the ability to rationalize a denominator with a higher level root
- Demonstrate the ability to rationalize a binomial denominator
#1:
Instructions: Simplify each.
Assume all variables are positive real numbers.
$$a)\hspace{.2em}\frac{2\sqrt{5}}{4\sqrt{2}}$$
$$b)\hspace{.2em}\frac{5\sqrt{2}}{5\sqrt{5}}$$
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#2:
Instructions: Simplify each.
Assume all variables are positive real numbers.
$$a)\hspace{.2em}\frac{4 - 4\sqrt{2x^3}}{4\sqrt{20x^3}}$$
$$b)\hspace{.2em}\frac{4 + 3\sqrt{2x^3}}{4\sqrt{3x}}$$
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#3:
Instructions: Simplify each.
Assume all variables are positive real numbers.
$$a)\hspace{.2em}\frac{2 \sqrt{3x^2y}- 3\sqrt{3x^3y^2}}{4\sqrt{13xy^3}}$$
$$b)\hspace{.2em}\frac{5x}{3\sqrt{3x^3}- 2\sqrt{3x}}$$
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#4:
Instructions: Simplify each.
Assume all variables are positive real numbers.
$$a)\hspace{.2em}\frac{4}{5\sqrt{5x^3}+ 3\sqrt{2x^3}}$$
$$b)\hspace{.2em}\frac{-4 + 5x}{5\sqrt[3]{12x^4}}$$
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#5:
Instructions: Simplify each.
Assume all variables are positive real numbers.
$$a)\hspace{.2em}\frac{-5 + 5 \sqrt[4]{x^3}}{4\sqrt[4]{12x^3}}$$
$$b)\hspace{.2em}\frac{-4 + 3x^2}{2x + 2\sqrt{3x}}$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}\frac{\sqrt{10}}{4}$$
$$b)\hspace{.2em}\frac{\sqrt{10}}{5}$$
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#2:
Solutions:
$$a)\hspace{.2em}\frac{\sqrt{5x}- x^2\sqrt{10}}{10x^2}$$
$$b)\hspace{.2em}\frac{4\sqrt{3x}+ 3x^2\sqrt{6}}{12x}$$
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#3:
Solutions:
$$a)\hspace{.2em}\frac{2\sqrt{39x}- 3x\sqrt{39y}}{52y}$$
$$b)\hspace{.2em}\frac{5\sqrt{3x}}{9x - 6}$$
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#4:
Solutions:
$$a)\hspace{.2em}\frac{20\sqrt{5x}- 12\sqrt{2x}}{107x^2}$$
$$b)\hspace{.2em}\frac{-4\sqrt[3]{18x^2}+ 5x\sqrt[3]{18x^2}}{30x^2}$$
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#5:
Solutions:
$$a)\hspace{.2em}\frac{-5\sqrt[4]{108x}+ 5x\sqrt[4]{108}}{24x}$$
$$b)\hspace{.2em}\frac{3x^3 - 3x^2\sqrt{3x}- 4x + 4\sqrt{3x}}{2x^2 - 6x}$$
