[latex]\\sqrt{x}=x^{\\frac{1}{2}}\\hspace{0.25in} \\sqrt[3]{x}=x^{\\frac{1}{3}}\\hspace{0.25in} \\sqrt[4]{x}=x^{\\frac{1}{4}}\\hspace{0.25in} \\sqrt[5]{x}=x^{\\frac{1}{5}}[\/latex]<\/p>\r\nRadicals having some exponent value inside the radical can also be written as a fractional exponent. For example:\r\n
[latex]\\sqrt{x^3}=x^{\\frac{3}{2}}\\hspace{0.25in} \\sqrt[3]{x^2}=x^{\\frac{2}{3}}\\hspace{0.25in} \\sqrt[4]{x^5}=x^{\\frac{5}{4}}\\hspace{0.25in} \\sqrt[5]{x^9}=x^{\\frac{9}{5}}[\/latex]<\/p>\r\nThe general form that radicals having exponents take is:\r\n
[latex]x^{\\frac{b}{a}}=\\sqrt[a]{x^b}\\text{ or }(\\sqrt[a]{x})^b[\/latex]<\/p>\r\nShould the reciprocal of a radical having an exponent, it would look as follows:\r\n
[latex]x^{-\\frac{b}{a}}=\\dfrac{1}{\\sqrt[a]{x^b}}\\text{ or }\\dfrac{1}{(\\sqrt[a]{x})^b}[\/latex]<\/p>\r\nIn both cases shown above, the power of the radical is [latex]b[\/latex] and the root of the radical is [latex]a[\/latex]. These are the two forms that a radical having an exponent is commonly written in. It is convenient to work with a radical containing an exponent in one of these two forms.\r\n
Example 9.6.1<\/p>\r\n\r\n<\/header>\r\n
[latex]\\dfrac{1}{\\sqrt[3]{27^4}}\\text{ or }\\dfrac{1}{(\\sqrt[3]{27})^4}[\/latex]<\/p>\r\nFirst, the cube root of 27 will reduce to 3, which leaves:\r\n
[latex]\\dfrac{1}{3^4}\\text{ or }\\dfrac{1}{81}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nOnce the radical having an exponent is converted into a pure fractional exponent, then the following rules can be used.\r\n
[latex]\\begin{array}{ccc}\r\na^ma^n=a^{m+n}\\hspace{0.25in} &(ab)^m=a^mb^m\\hspace{0.25in} &a^{-m}=\\dfrac{1}{a^m} \\\\ \\\\\r\n\\dfrac{a^m}{a^n}=a^{m-n}&\\left(\\dfrac{a}{b}\\right)=\\dfrac{a^m}{b^m}&\\dfrac{1}{a^{-m}}=a^m \\\\ \\\\\r\n(a^m)^n=a^{mn}&a^0=1&\\left(\\dfrac{a}{b}\\right)^{-m}=\\dfrac{b^m}{a^m}\r\n\\end{array}[\/latex]<\/p>\r\n\r\n
Example 9.6.2<\/p>\r\n\r\n<\/header>\r\n
[latex](x^2y^{\\frac{4}{3}})(x^{-1}y^\\frac{2}{3})[\/latex] becomes [latex]x^2\\cdot x^{-1}\\cdot y^{\\frac{4}{3}}\\cdot y^{\\frac{2}{3}}[\/latex]<\/p>\r\nCombining the exponents yields:\r\n
[latex]x^{2 - 1}\\cdot y^{\\frac{4}{3}+\\frac{2}{3}}[\/latex]<\/p>\r\nWhich results in:\r\n
[latex]x^1\\cdot y^{\\frac{6}{3}}[\/latex]<\/p>\r\nWhich simplifies to:\r\n
[latex]xy^2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n
Example 9.6.3<\/p>\r\n\r\n<\/header>\r\n
[latex]\\dfrac{ab^{\\frac{2}{3}}3b^{-\\frac{5}{3}}}{5a^{-\\frac{3}{2}}b^{-\\frac{4}{3}}}[\/latex] becomes [latex]3\\cdot 5^{-1}\\cdot a \\cdot a^{\\frac{3}{2}}\\cdot b^{\\frac{2}{3}}\\cdot b^{-\\frac{5}{3}}\\cdot b^{\\frac{4}{3}}[\/latex][footnote]When we divide by an exponent, we subtract powers.[\/footnote]<\/p>\r\nCombining the exponents yields:\r\n
[latex]3\\cdot 5^{-1}\\cdot a^{1+\\frac{3}{2}}\\cdot b^{\\frac{2}{3}-\\frac{5}{3}+\\frac{4}{3}}[\/latex]<\/p>\r\n
Which gives:<\/p>\r\n
[latex]3\\cdot 5^{-1}\\cdot a^{\\frac{5}{2}}\\cdot b^{\\frac{1}{3}}[\/latex]<\/p>\r\n
Which simplifies to:<\/p>\r\n
[latex]\\dfrac{3\\cdot a^{\\frac{5}{2}}\\cdot b^{\\frac{1}{3}}}{5}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n
When simplifying radicals that use fractional exponents, the numerator on the exponent is divided by the denominator. All radicals can be shown as having an equivalent fractional exponent. For example:<\/p>\n
[latex]\\sqrt{x}=x^{\\frac{1}{2}}\\hspace{0.25in} \\sqrt[3]{x}=x^{\\frac{1}{3}}\\hspace{0.25in} \\sqrt[4]{x}=x^{\\frac{1}{4}}\\hspace{0.25in} \\sqrt[5]{x}=x^{\\frac{1}{5}}[\/latex]<\/p>\n
Radicals having some exponent value inside the radical can also be written as a fractional exponent. For example:<\/p>\n
[latex]\\sqrt{x^3}=x^{\\frac{3}{2}}\\hspace{0.25in} \\sqrt[3]{x^2}=x^{\\frac{2}{3}}\\hspace{0.25in} \\sqrt[4]{x^5}=x^{\\frac{5}{4}}\\hspace{0.25in} \\sqrt[5]{x^9}=x^{\\frac{9}{5}}[\/latex]<\/p>\n
The general form that radicals having exponents take is:<\/p>\n
[latex]x^{\\frac{b}{a}}=\\sqrt[a]{x^b}\\text{ or }(\\sqrt[a]{x})^b[\/latex]<\/p>\n
Should the reciprocal of a radical having an exponent, it would look as follows:<\/p>\n
[latex]x^{-\\frac{b}{a}}=\\dfrac{1}{\\sqrt[a]{x^b}}\\text{ or }\\dfrac{1}{(\\sqrt[a]{x})^b}[\/latex]<\/p>\n
In both cases shown above, the power of the radical is [latex]b[\/latex] and the root of the radical is [latex]a[\/latex]. These are the two forms that a radical having an exponent is commonly written in. It is convenient to work with a radical containing an exponent in one of these two forms.<\/p>\n
Example 9.6.1<\/p>\n<\/header>\n
Evaluate [latex]27^{-\\frac{4}{3}}[\/latex].<\/p>\n
Converting to a radical form:<\/p>\n
[latex]\\dfrac{1}{\\sqrt[3]{27^4}}\\text{ or }\\dfrac{1}{(\\sqrt[3]{27})^4}[\/latex]<\/p>\n
First, the cube root of 27 will reduce to 3, which leaves:<\/p>\n
[latex]\\dfrac{1}{3^4}\\text{ or }\\dfrac{1}{81}[\/latex]<\/p>\n<\/div>\n<\/div>\n
Once the radical having an exponent is converted into a pure fractional exponent, then the following rules can be used.<\/p>\n
[latex]\\begin{array}{ccc} a^ma^n=a^{m+n}\\hspace{0.25in} &(ab)^m=a^mb^m\\hspace{0.25in} &a^{-m}=\\dfrac{1}{a^m} \\\\ \\\\ \\dfrac{a^m}{a^n}=a^{m-n}&\\left(\\dfrac{a}{b}\\right)=\\dfrac{a^m}{b^m}&\\dfrac{1}{a^{-m}}=a^m \\\\ \\\\ (a^m)^n=a^{mn}&a^0=1&\\left(\\dfrac{a}{b}\\right)^{-m}=\\dfrac{b^m}{a^m} \\end{array}[\/latex]<\/p>\n
Example 9.6.2<\/p>\n<\/header>\n
Simplify [latex](x^2y^{\\frac{4}{3}})(x^{-1}y^\\frac{2}{3})[\/latex].<\/p>\n
First, you need to separate the different variables:<\/p>\n
[latex](x^2y^{\\frac{4}{3}})(x^{-1}y^\\frac{2}{3})[\/latex] becomes [latex]x^2\\cdot x^{-1}\\cdot y^{\\frac{4}{3}}\\cdot y^{\\frac{2}{3}}[\/latex]<\/p>\n
Combining the exponents yields:<\/p>\n
[latex]x^{2 - 1}\\cdot y^{\\frac{4}{3}+\\frac{2}{3}}[\/latex]<\/p>\n
Which results in:<\/p>\n
[latex]x^1\\cdot y^{\\frac{6}{3}}[\/latex]<\/p>\n
Which simplifies to:<\/p>\n
[latex]xy^2[\/latex]<\/p>\n<\/div>\n<\/div>\n
Example 9.6.3<\/p>\n<\/header>\n
Simplify [latex]\\dfrac{ab^{\\frac{2}{3}}3b^{-\\frac{5}{3}}}{5a^{-\\frac{3}{2}}b^{-\\frac{4}{3}}}[\/latex].<\/p>\n
First, separate the different variables:<\/p>\n
[latex]\\dfrac{ab^{\\frac{2}{3}}3b^{-\\frac{5}{3}}}{5a^{-\\frac{3}{2}}b^{-\\frac{4}{3}}}[\/latex] becomes [latex]3\\cdot 5^{-1}\\cdot a \\cdot a^{\\frac{3}{2}}\\cdot b^{\\frac{2}{3}}\\cdot b^{-\\frac{5}{3}}\\cdot b^{\\frac{4}{3}}[\/latex][1]<\/sup><\/a><\/p>\n Combining the exponents yields:<\/p>\n [latex]3\\cdot 5^{-1}\\cdot a^{1+\\frac{3}{2}}\\cdot b^{\\frac{2}{3}-\\frac{5}{3}+\\frac{4}{3}}[\/latex]<\/p>\n Which gives:<\/p>\n [latex]3\\cdot 5^{-1}\\cdot a^{\\frac{5}{2}}\\cdot b^{\\frac{1}{3}}[\/latex]<\/p>\n Which simplifies to:<\/p>\n [latex]\\dfrac{3\\cdot a^{\\frac{5}{2}}\\cdot b^{\\frac{1}{3}}}{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n Write each of the following fractional exponents in radical form.<\/p>\n Write each of the following radicals in exponential form.<\/p>\n Evaluate the following.<\/p>\n Simplify. Your answer should only contain positive exponents.<\/p>\nQuestions<\/h1>\n
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