Factoring Radicals - Math

Card 0 of 76

Question

Factor and simplify the following radical expression:

$$\frac{2-\sqrt{2}$}{4+2$\sqrt{2}$}$

Answer

Begin by multiplying the numerator and denominator by the complement of the denominator:

$$\frac{2-\sqrt{2}$}{4+2$\sqrt{2}$}$

$$\frac{2-\sqrt{2}$}{4+2$\sqrt{2}$} \cdot $\frac{4-2\sqrt{2}$}{4-2$\sqrt{2}$}$

$$\frac{8-4\sqrt{2}$-4$\sqrt{2}$+4}{16-8}$

Combine like terms and simplify:

$$\frac{12-8\sqrt{2}$}{8}$

$$\frac{3-2\sqrt{2}$}{2}$

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Question

Find the value of $\sqrt{80}$ + $\sqrt{20}$ .

Answer

To solve this equation, we have to factor our radicals. We do this by finding numbers that multiply to give us the number within the radical.

$\sqrt{80}$ = $\sqrt{20}$$\sqrt{4}$

$\sqrt{20}$ = $\sqrt{5}$$\sqrt{4}$

Add them together:

$\sqrt5\sqrt4\sqrt4 + \sqrt5\sqrt4$

4 is a perfect square, so we can find the root:

$(2)(2)\sqrt5 + 2\sqrt5$

$4\sqrt5 + 2\sqrt5 = 6\sqrt5$

Since both have the same radical, we can combine them:

$4\sqrt5 + 2\sqrt5 = 6\sqrt5$

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Question

Simplify the expression:

$$\frac{3\sqrt[4]{32}$}{2\sqrt[4]{162}}$

Answer

Use the multiplication property of radicals to split the fourth roots as follows:

$\rightarrow $\frac{3\sqrt[4]{16}$\sqrt[4]{2}}{2\sqrt[4]{81}\sqrt[4]{2}}$

Simplify the new roots:

$\rightarrow $\frac{3(2)\sqrt[4]{2}$}{2(3)\sqrt[4]{2}}$

$\rightarrow $\frac{6\sqrt[4]{2}$}{6\sqrt[4]{2}}$

$\rightarrow 1$

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Question

Factor and simplify the following radical expression:

Answer

Begin by factoring the integer:

Now, simplify the exponents:

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Question

Factor and simplify the following radical expression:

Answer

Begin by converting the radical into exponent form:

Now, multiply the exponents:

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Question

Factor and simplify the following radical expression:

Answer

Begin by converting the radical into exponent form:

Now, combine the bases:

Simplify the integer:

$(2\cdot 3\cdot $3^4a$^$5b^7c$^$6)^{$\frac{1}${4}$}$

Now, simplify the exponents:

$(2\cdot 3\cdot $3^4aa$^$4b^3b$^$4c^2c$^$4)^{$\frac{1}${4}$}$

Convert back into radical form and simplify:

$\sqrt[4]{2\cdot 3\cdot $3^4aa$^$4b^3b$^$4c^2c$^4}$

$3abc\sqrt[4]{2\cdot $3ab^3c$^2}$

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Question

Factor and simplify the following radical expression:

$(4-2$\sqrt{2}$)($\sqrt{2}$-4)$

Answer

Begin by using the FOIL method (First Outer Inner Last) to expand the expression.

$(4-2$\sqrt{2}$)($\sqrt{2}$-4)$

$4$\sqrt{2}$-16-4+8$\sqrt{2}$$

Now, combine like terms:

$12$\sqrt{2}$-20$

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Question

Factor and simplify the following radical expression:

$$\frac{3-2\sqrt{3}$}{2$\sqrt{3}$+3}$

Answer

Begin by multiplying the numerator and denominator by the complement of the denominator:

$$\frac{3-2\sqrt{3}$}{2$\sqrt{3}$+3} \cdot $\frac{2\sqrt{3}$-3}{2$\sqrt{3}$-3}$

Use the FOIL method to multiply the radicals. F (first) O (outer) I (inner) L (last)

$$\frac{6\sqrt{3}$-9-12+6$\sqrt{3}$}{12-6$\sqrt{3}$+6$\sqrt{3}$-9}$

Now, combine like terms:

$$\frac{12\sqrt{3}$-21}{3}$

Simplify:

$4$\sqrt{3}$-7$

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Question

Factor and simplify the following radical expression:

$\sqrt{\frac{1}{5}$}-2$\sqrt{20}$+3$\sqrt{5}$

Answer

Begin by factoring the radicals:

$\sqrt{\frac{1}{5}$}-2$\sqrt{20}$+3$\sqrt{5}$

$\sqrt{\frac{1}{5}$}-4$\sqrt{5}$+3$\sqrt{5}$

Combine like terms:

$\sqrt{\frac{1}{5}$}-$\sqrt{5}$

Multiply the left side by $$\frac{\sqrt{5}$}{$\sqrt{5}$}$ and the right side by $\frac{5}{5}$

$\frac{1}{\sqrt{5}$}-$\sqrt{5}$

$\frac{1}{\sqrt{5}$}\cdot $\frac{\sqrt{5}$}{$\sqrt{5}$}-$\sqrt{5}$\cdot$\frac{5}{5}$

$$\frac{\sqrt{5}$}{5}-$\frac{5\sqrt{5}$}{5}$

$$\frac{-4\sqrt{5}$}{5}$

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Question

Factor and simplify the following radical expression:

Answer

Begin by converting the radicals into exponent form:

$(3\cdot $2^3a$^$2b^3c$^$5)^{$\frac{1}${4}$}\cdot(3\cdot $2^4a$^$3b^3c$^$6)^{$\frac{1}${4}$}$

Now, combine the bases:

Convert back into radical form and simplify:

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Question

Factor and simplify the following radical expression:

Answer

Begin by simplifying the right side of the rational expression:

$\sqrt{3mn}$+3m$\sqrt{3mn}$

$1$\sqrt{3mn}$+3m$\sqrt{3mn}$$

Now, combine like terms:

$(1+3m)$\sqrt{3mn}$$

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Question

Factor and simplify the following radical expression:

$(4-$\sqrt{3}$)(6-$\sqrt{3}$)$

Answer

Begin by using the FOIL method to multiply the radical expression. F (first) O (outer) I (inner) L (last)

$(4-$\sqrt{3}$)(6-$\sqrt{3}$)$

$24-4$\sqrt{3}$-6$\sqrt{3}$+3$

Now, combine like terms:

$27-10$\sqrt{3}$$

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Question

Factor and simplify the following radical expression:

$\sqrt[3]{$\frac{2}{5n}$}$

Answer

Begin by multiplying the numerator and denominator by :

$\sqrt[3]{$\frac{2}{5n}$}$

$$\frac{\sqrt[3]{2}$}{\sqrt[3]{5n}}\cdot$

The expression cannot be further simplified.

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Question

Factor and simplify the following radical expression:

$\sqrt{\frac{1}{3}$}+$\sqrt{75}$-2$\sqrt{3}$

Answer

Begin by factoring the radicals and combining like terms:

$\sqrt{\frac{1}{3}$}+$\sqrt{75}$-2$\sqrt{3}$

$\sqrt{\frac{1}{3}$}+5$\sqrt{3}$-2$\sqrt{3}$

$\sqrt{\frac{1}{3}$}+3$\sqrt{3}$

Multiply the left side of the equation by $$\frac{\sqrt{3}$}{$\sqrt{3}$}$ and the right side by $\frac{3}{3}$ :

${$\frac{1}{\sqrt{3}$}}+3$\sqrt{3}$$

${$\frac{1}{\sqrt{3}$}}\cdot $\frac{\sqrt{3}$}{$\sqrt{3}$}+3$\sqrt{3}$\cdot$\frac{3}{3}$$

$$\frac{\sqrt{3}$}{3}+\frac{9\sqrt{3}$}{3}$

$$\frac{10\sqrt{3}$}{3}$

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Question

Simplify the following radical expression:

Answer

Begin by factoring the integer:

Factor the exponents:

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Question

Simplify the following radical expression:

Answer

Begin by factoring the integer:

Factor the exponents:

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Question

Simplify the following radical expression:

Answer

Begin by factoring the integer:

Factor the exponents:

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Question

Simplify the following radical expression:

Answer

Begin by factoring the expression:

Now, take the square root:

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Question

Simplify the following radical expression:

Answer

Simplify the radical expression:

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Question

Factor and simplify the following radical expression:

$$\frac{2-\sqrt{2}$}{4+2$\sqrt{2}$}$

Answer

Begin by multiplying the numerator and denominator by the complement of the denominator:

$$\frac{2-\sqrt{2}$}{4+2$\sqrt{2}$}$

$$\frac{2-\sqrt{2}$}{4+2$\sqrt{2}$} \cdot $\frac{4-2\sqrt{2}$}{4-2$\sqrt{2}$}$

$$\frac{8-4\sqrt{2}$-4$\sqrt{2}$+4}{16-8}$

Combine like terms and simplify:

$$\frac{12-8\sqrt{2}$}{8}$

$$\frac{3-2\sqrt{2}$}{2}$

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