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10.9: Radicals- Answers to the Homework Exercises

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    Simplify Radicals

    1. \(7\sqrt{5}\)
    1. \(2\sqrt{3}\)
    1. \(48\sqrt{2}\)
    1. \(8\sqrt{3n}\)
    1. \(6x\sqrt{7}\)
    1. \(-56x^2\)
    1. \(3xy\sqrt{5}\)
    1. \(8x^2y^2\sqrt{5}\)
    1. \(35xy\sqrt{5y}\)
    1. \(-48x^2z^2y\sqrt{5}\)
    1. \(-12p\sqrt{6mn}\)
    1. \(14\)
    1. \(20\sqrt{2}\)
    1. \(-21\sqrt{7}\)
    1. \(10n\sqrt{n}\)
    1. \(-20p^2\sqrt{7}\)
    1. \(32p\sqrt{7}\)
    1. \(16a^2b\sqrt{2}\)
    1. \(56\sqrt{2mn}\)
    1. \(-30y^2x\sqrt{2x}\)
    1. \(-4yz\sqrt{2xz}\)
    1. \((\sqrt[5]{m})^3\)
    1. \((\sqrt{7x})^3\)
    1. \((6x)^{-\dfrac{3}{2}}\)
    1. \(n^{-\dfrac{7}{4}}\)
    1. \(4\)
    1. \(8\)

    Add and Subtract Radicals

    1. \(6\sqrt{5}\)
    1. \(-5\sqrt{6}\)
    1. \(-8\sqrt{2}\)
    1. \(-2\sqrt{2}\)
    1. \(-3\sqrt{6}-\sqrt{3}\)
    1. \(-4\sqrt{6}+4\sqrt{5}\)
    1. \(2\sqrt[3]{2}\)
    1. \(\sqrt[4]{2}-3\sqrt[4]{3}\)
    1. \(2\sqrt[4]{2}+\sqrt[4]{3}+6\sqrt[4]{4}\)
    1. \(4\sqrt[5]{5}-4\sqrt[5]{6}\)
    1. \(-3\sqrt{6}-5\sqrt{3}\)
    1. \(-3\sqrt{3}\)
    1. \(-6\sqrt{6}+9\sqrt{3}\)
    1. \(8\sqrt{5}-\sqrt{3}\)
    1. \(3\sqrt{2}+3\sqrt{6}\)
    1. \(-\sqrt{5}-3\sqrt{6}\)
    1. \(6\sqrt[3]{5}-3\sqrt[3]{3}\)
    1. \(5\sqrt[4]{6}+2\sqrt[4]{4}\)
    1. \(-2\sqrt[4]{3}-9\sqrt[4]{5}-3\sqrt[4]{2}\)
    1. \(-11\sqrt[7]{2}-2\sqrt[7]{5}\)

    Multiply and Divide Radicals

    1. \(-48\sqrt{5}\)
    1. \(2x^2\sqrt[3]{x}\)
    1. \(-45\sqrt{5}-10\sqrt{15}\)
    1. \(-2-4\sqrt{2}\)
    1. \(6a+a\sqrt{10}+6a\sqrt{6}+2a\sqrt{15}\)
    1. \(\dfrac{\sqrt{3}}{25}\)
    1. \(\dfrac{\sqrt{5}}{2}\)
    1. \(\dfrac{5}{12y^4}\)
    1. \(\dfrac{\sqrt[3]{10}}{5}\)
    1. \(\dfrac{5\sqrt[4]{r^2}}{2}\) or \(\dfrac{5\sqrt{r}}{2}\)
    1. \(-25r^2\sqrt{2r}\)
    1. \(5\sqrt{2}+2\sqrt{5}\)
    1. \(5\sqrt{3}-9\sqrt{5v}\)
    1. \(30+8\sqrt{3}+5\sqrt{15}+4\sqrt{5}\)
    1. \(-10\sqrt{m}+25\sqrt{2}+\sqrt{2m}-5\)
    1. \(2\)
    1. \(4\sqrt{2}\)
    1. \(\dfrac{\sqrt{n}}{2}\)
    1. \(\dfrac{1}{4}\)

    Rationalize Denominators

    1. \(\dfrac{4\sqrt{3}}{9}\)
    1. \(2\)
    1. \(\dfrac{4\sqrt{5}}{5}\)
    1. \(\dfrac{2+\sqrt{3}}{5}\)
    1. \(\dfrac{\sqrt{6}-9}{3}\)
    1. \(\dfrac{10-2\sqrt{2}}{23}\)
    1. \(3-\sqrt{5}\)
    1. \(\sqrt{2}-1\)
    1. \(\sqrt{a}\)
    1. \(4-2\sqrt{3}+2\sqrt{6}-3\sqrt{2}\)
    1. \(3\sqrt{2}+2\sqrt{3}\)
    1. \(\dfrac{-1+\sqrt{5}}{4}\)
    1. \(\dfrac{\sqrt{3}-1}{4}\)
    1. \(\dfrac{\sqrt{30}-2\sqrt{3}}{18}\)
    1. \(\dfrac{2\sqrt{3}+\sqrt{2}}{2}\)
    1. \(\dfrac{\sqrt{5}-\sqrt{3}}{2}\)
    1. \(3+2\sqrt{3}\)
    1. \(3-2\sqrt{2}\)
    1. \(\dfrac{2\sqrt{5}-2\sqrt{15}+\sqrt{3}+3}{-2}\)
    1. \(\dfrac{a\sqrt{b}+b\sqrt{a}}{a-b}\)
    1. \(\dfrac{2\sqrt{5}-5\sqrt{2}-10+5\sqrt{10}}{30}\)
    1. \(\dfrac{8+3\sqrt{6}}{10}\)

    Radicals with Mixed Indices

    1. \(\sqrt[4]{4x^2y^3}\)
    1. \(\dfrac{\sqrt[3]{36xy}}{3y}\)
    1. \(\sqrt[4]{x^3y^2z}\)
    1. \(\sqrt{3xy^3}\)
    1. \(\sqrt[5]{x^3y^4z^2}\)
    1. \(\sqrt{5y}\)
    1. \(\sqrt[6]{5400}\)
    1. \(\sqrt[6]{x^3(x-2)^2}\)
    1. \(\sqrt[12]{x^{11}y^{10}}\)
    1. \(a\sqrt[4]{a}\)
    1. \(xy\sqrt[6]{xy^5}\)
    1. \(x\sqrt[12]{59049xy^{11}z^{10}}\)
    1. \(\sqrt[12]{a^5}\)
    1. \(\sqrt[10]{ab^9c^7}\)
    1. \(\sqrt[15]{(2x+1)^4}\)
    1. \(\sqrt[15]{27y^5z^5}\)
    1. \(\sqrt[10]{4a^9b^9}\)
    1. \(\sqrt[30]{x^{22}y^{11}z^{27}}\)
    1. \(a\sqrt[12]{a^5}\)
    1. \(2xy^2\sqrt[6]{2x^5y}\)
    1. \(4x(y+z)^3\sqrt[6]{2x(y+z)}\)
    1. \(\dfrac{\sqrt[15]{a^7b^{11}}}{b}\)
    1. \(\sqrt[12]{(2+5x)^5}\)

    Radical Equations

    1. \(3\)
    1. \(\pm 2\)
    1. \(5\)
    1. \(5\)
    1. \(3\)
    1. \(3\)
    1. \(7\)
    1. \(21\)
      1. \(79.5\) inches
      2. \(145.7\) pounds

    Solving with Rational Exponents

    1. \(\pm 5\sqrt{3}\)
    1. \(\pm 2\sqrt{6}\)
    1. \(-1\)
    1. \(-7\)
    1. \(-\dfrac{3}{8}, -\dfrac{5}{8}\)
    1. \(-\dfrac{34}{3}, -10\)
    1. \(-2\)
    1. \(-3,11\)
    1. \(\dfrac{-1\pm 3\sqrt{2}}{2}\)
    1. \(-\dfrac{11}{2},\dfrac{5}{2}\)
    1. \(\dfrac{9}{8}\)
    1. \(3\)

    Complex Numbers

    1. \(8i\)
    1. \(9i\)
    1. \(10i\)
    1. \(3i\sqrt{10}\)
    1. \(7i\sqrt{5}\)
    1. \(40\)
    1. \(-12\)
    1. \(-15\)
    1. \(-2\sqrt{5}\)
    1. \(-7-4i\)
    1. \(2-5i\)
    1. \(\dfrac{1+i\sqrt{3}}{2}\)
    1. \(\dfrac{2+i\sqrt{2}}{2}\)
    1. \(5-i\sqrt{3}\)
    1. \(\dfrac{5+2i\sqrt{3}}{2}\)
    1. \(11-4i\)
    1. \(-3-13i\)
    1. \(-8-2i\)
    1. \(80-10i\)
    1. \(44+8i\)
    1. \(9i+5\)
    1. \(\dfrac{3i-6}{4}\)
    1. \(\dfrac{-40i+4}{101}\)
    1. \(\dfrac{70+49i}{149}\)
    1. \(-4i\)
    1. \(5-12i\)
    1. \(13-8i\)
    1. \(-32-128i\)
    1. \(-28+76i\)
    1. \(-1+13i\)
    1. \(\dfrac{4i+2}{3}\)
    1. \(-2i\)
    1. \(\dfrac{4-6i}{13}\)
    1. \(\dfrac{48i-56}{85}\)
    1. \(1\)
    1. \(-1\)
    1. \(1\)
    1. \(-i\)

    This page titled 10.9: Radicals- Answers to the Homework Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Darlene Diaz (ASCCC Open Educational Resources Initiative) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.