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- https://math.libretexts.org/Courses/Highline_College/MATHP_141%3A_Corequisite_Precalculus/02%3A_Algebra_Support/2.19%3A_Simplifying_Rational_ExponentsRemember the Power Property tells us to multiply the exponents and so \left(a^{\frac{1}{n}}\right)^{m} and \left(a^{m}\right)^{\frac{1}{n}} both equal a^{\frac{m}{n}}. \(a^{\frac{m}{n}}=\l...Remember the Power Property tells us to multiply the exponents and so \left(a^{\frac{1}{n}}\right)^{m} and \left(a^{m}\right)^{\frac{1}{n}} both equal a^{\frac{m}{n}}. a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^{m}=(\sqrt[n]{a})^{m} \quad \text { and } \quad a^{\frac{m}{n}}=\left(a^{m}\right)^{^{\frac{1}{n}}}=\sqrt[n]{a^{m}} We want to use a^{\frac{m}{n}}=\sqrt[n]{a^{m}} to write each radical in the form a^{\frac{m}{n}}
- https://math.libretexts.org/Courses/Coastline_College/Math_C045%3A_Beginning_and_Intermediate_Algebra_(Tran)/10%3A_Roots_and_Radicals/10.04%3A_Simplify_Rational_ExponentsRational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.
- https://math.libretexts.org/Courses/Kansas_State_University/Your_Guide_to_Intermediate_Algebra/05%3A_Everything_else_you_need_to_know/5.01%3A_Simplify_Rational_ExponentsWe notice that if a>0, that is a positive real number, then \sqrt[n]{a^{n}}=a for n\geq 2 no matter if n is even or odd. Say we have a fraction \dfrac{1}{\sqrt{2}} and we want to make ...We notice that if a>0, that is a positive real number, then \sqrt[n]{a^{n}}=a for n\geq 2 no matter if n is even or odd. Say we have a fraction \dfrac{1}{\sqrt{2}} and we want to make the denominator into a rational, that is we want to rationalize the denominator. So we get (a^n)^m=a^{n\cdot m}=a^{m\cdot n}=(a^m)^n. The same thing goes for to the power of 3, (a\cdot b)^3=a\cdot b \cdot a\cdot b \cdot a\cdot b=a\cdot a \cdot a\cdot b \cdot b\cdot b=a^3\cdot b^3.
- https://math.libretexts.org/Courses/City_University_of_New_York/MAT1275_Basic/06%3A_Roots_and_Radicals/6.03%3A_Simplify_Rational_Exponentsa^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^{m}=(\sqrt[n]{a})^{m} \quad \text { and } \quad a^{\frac{m}{n}}=\left(a^{m}\right)^{^{\frac{1}{n}}}=\sqrt[n]{a^{m}} a. \(\dfrac{a^{\frac{3}{4}} \cdot a...a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^{m}=(\sqrt[n]{a})^{m} \quad \text { and } \quad a^{\frac{m}{n}}=\left(a^{m}\right)^{^{\frac{1}{n}}}=\sqrt[n]{a^{m}} a. \dfrac{a^{\frac{3}{4}} \cdot a^{-\frac{1}{4}}}{a^{-\frac{10}{4}}} b. \left(\dfrac{27 b^{\frac{2}{3}} c^{-\frac{5}{2}}}{b^{-\frac{7}{3}} c^{\frac{1}{2}}}\right)^{\frac{1}{3}}
- https://math.libretexts.org/Courses/Las_Positas_College/Foundational_Mathematics/17%3A_Radical_Expressions_and_Functions/17.04%3A_Simplify_Rational_ExponentsRational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.
- https://math.libretexts.org/Courses/Nova_Scotia_Community_College/MATH_1043/01%3A_Numerical_Literacy/1.05%3A_Expressions/1.5.04%3A_Unit_1_-_Chapter_6-_Roots_and_Radicals/1.5.4.03%3A_Simplify_Rational_ExponentsRational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.
- https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT_206.5/Chapter_3A%3A_Algebra_Topics/3A.10%3A_Rational_ExponentsRational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.
- https://math.libretexts.org/Courses/City_University_of_New_York/College_Algebra_and_Trigonometry-_Expressions_Equations_and_Graphs/01%3A_Expressions/1.04%3A_Radical_Expressions/1.4.03%3A_Rational_ExponentsRemember the Power Property tells us to multiply the exponents and so \left(a^{\frac{1}{n}}\right)^{m} and \left(a^{m}\right)^{\frac{1}{n}} both equal a^{\frac{m}{n}}. \(a^{\frac{m}{n}}=\l...Remember the Power Property tells us to multiply the exponents and so \left(a^{\frac{1}{n}}\right)^{m} and \left(a^{m}\right)^{\frac{1}{n}} both equal a^{\frac{m}{n}}. a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^{m}=(\sqrt[n]{a})^{m} \quad \text { and } \quad a^{\frac{m}{n}}=\left(a^{m}\right)^{^{\frac{1}{n}}}=\sqrt[n]{a^{m}} Give and example of the rules (ab)^n=a^nb^n and a^na^m=a^{n+m} with rational exponents.
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_104_Intermediate_Algebra/8%3A_Roots_and_Radicals/8.2%3A_Simplify_Rational_ExponentsRational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.
- https://math.libretexts.org/Courses/Coastline_College/Math_C097%3A_Support_for_Precalculus_Corequisite%3A_MATH_C170/1.02%3A_Algebra_Support/1.2.19%3A_Simplifying_Rational_ExponentsRemember the Power Property tells us to multiply the exponents and so \left(a^{\frac{1}{n}}\right)^{m} and \left(a^{m}\right)^{\frac{1}{n}} both equal a^{\frac{m}{n}}. \(a^{\frac{m}{n}}=\l...Remember the Power Property tells us to multiply the exponents and so \left(a^{\frac{1}{n}}\right)^{m} and \left(a^{m}\right)^{\frac{1}{n}} both equal a^{\frac{m}{n}}. a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^{m}=(\sqrt[n]{a})^{m} \quad \text { and } \quad a^{\frac{m}{n}}=\left(a^{m}\right)^{^{\frac{1}{n}}}=\sqrt[n]{a^{m}} We want to use a^{\frac{m}{n}}=\sqrt[n]{a^{m}} to write each radical in the form a^{\frac{m}{n}}
- https://math.libretexts.org/Courses/Highline_College/Math_098%3A_Intermediate_Algebra_for_Calculus/04%3A_Chapter_4_-_Radicals/4.03%3A_Simplify_Rational_ExponentsRational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.
