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:

$\sqrt[4]{81x^6y^5z^{10}}$

Answer

Begin by factoring the integer:

$\sqrt[4]{81x^6y^5z^{10}}$

$\sqrt[4]{3^4x^6y^5z^{10}}$

$3\sqrt[4]{x^6y^5z^{10}}$

Now, simplify the exponents:

$3\sqrt[4]{x^4x^2y^4yz^{4}z^4z^2}$

$3xyzz\sqrt[4]{x^2yz^2}$

$3xyz^2\sqrt[4]{x^2yz^2}$

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Question

Factor and simplify the following radical expression:

$\sqrt[3]{(3n-2)^3}$

Answer

Begin by converting the radical into exponent form:

$\sqrt[3]{(3n-2)^3}$

$(3n-2)^3^{(\frac{1}{3})}$

Now, multiply the exponents:

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Question

Factor and simplify the following radical expression:

$\sqrt[4]{18a^3bc^5}\cdot \sqrt[4]{27a^2b^6c}$

Answer

Begin by converting the radical into exponent form:

$\sqrt[4]{18a^3bc^5}\cdot \sqrt[4]{27a^2b^6c}$

$(18a^3bc^5)^{\frac{1}{4}}\cdot ({27a^2b^6c})^{\frac{1}{4}}$

Now, combine the bases:

$(486a^5b^7c^6)^{\frac{1}{4}}$

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:

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

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

$3abc\sqrt[4]{6ab^3c^2}$

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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:

$\sqrt[4]{24a^2b^3c^5}\cdot\sqrt[4]{48a^3b^3c^6}$

Answer

Begin by converting the radicals into exponent form:

$\sqrt[4]{24a^2b^3c^5}\cdot\sqrt[4]{48a^3b^3c^6}$

$(24a^2b^3c^5)^{\frac{1}{4}}\cdot(48a^3b^3c^6)^{\frac{1}{4}}$

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

Now, combine the bases:

$(3^2\cdot 2^7a^5b^6c^{11})^{\frac{1}{4}}$

$(3^2\cdot 2^7aa^4b^2b^4c^3c^{8})^{\frac{1}{4}}$

Convert back into radical form and simplify:

$\sqrt[4]{3^2\cdot 2^7aa^4b^2b^4c^3c^{8}}$

$2abc^2\sqrt[4]{3^22^3ab^2c^3}$

$2abc^2\sqrt[4]{72ab^2c^3}$

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Question

Factor and simplify the following radical expression:

$\sqrt{3mn}+\sqrt{27m^3n}$

Answer

Begin by simplifying the right side of the rational expression:

$\sqrt{3mn}+\sqrt{27m^3n}$

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

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

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 $\frac{\sqrt[3]{(5n)^2}}{\sqrt[3]{(5n)^2}}$ :

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

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

$\frac{\sqrt[3]{50n^2}}{5n}$

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:

$\sqrt[4]{64x^5y^6z^9}$

Answer

Begin by factoring the integer:

$\sqrt[4]{64x^5y^6z^9}$

$\sqrt[4]{2^6x^5y^6z^9}$

Factor the exponents:

$4xyz^2\sqrt[4]{2xy^2z}$

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Question

Simplify the following radical expression:

$\sqrt{72a^5b^7c^6}$

Answer

Begin by factoring the integer:

$\sqrt{72a^5b^7c^6}$

$\sqrt{2^33^2a^5b^7c^6}$

Factor the exponents:

$6a^2b^3c^3\sqrt{2ab}$

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Question

Simplify the following radical expression:

$\sqrt[3]{-40n^7m^4}$

Answer

Begin by factoring the integer:

$\sqrt[3]{-40n^7m^4}$

$\sqrt[3]{-2^35n^7m^4}$

Factor the exponents:

$-2n^2m\sqrt[3]{5nm^}$

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Question

Simplify the following radical expression:

$\sqrt{n^2-10n+25}$

Answer

Begin by factoring the expression:

$\sqrt{n^2-10n+25}$

$\sqrt{(n-5)^2}$

Now, take the square root:

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Question

Simplify the following radical expression:

$\sqrt[3]{(3x-y)^5}$

Answer

Simplify the radical expression:

$\sqrt[3]{(3x-y)^5}$

$\sqrt[3]{(3x-y)^2(3x-y)^3}$

$(3x-y)\sqrt[3]{(3x-y)^2}$

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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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