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- https://math.libretexts.org/Courses/Las_Positas_College/Foundational_Mathematics/13%3A_Additional_Foundational_Content/13.08%3A_Roots_and_Radicals/13.8.08%3A_Rational_Exponents\[\begin{array}{ll} {\textbf{Product Property}}&{a^m·a^n=a^{m+n}}\\ {\textbf{Power Property}}&{(a^m)^n=a^{m·n}}\\ {\textbf{Product to a Power}}&{(ab)^m=a^{m}b^{m}}\\ {\textbf{Quotient Property}}&{\fra...\[\begin{array}{ll} {\textbf{Product Property}}&{a^m·a^n=a^{m+n}}\\ {\textbf{Power Property}}&{(a^m)^n=a^{m·n}}\\ {\textbf{Product to a Power}}&{(ab)^m=a^{m}b^{m}}\\ {\textbf{Quotient Property}}&{\frac{a^m}{a^n}=a^{m−n} , a \ne 0, m>n}\\ {}&{\frac{a^m}{a^n}=\frac{1}{a^{n−m}}, a \ne 0, n>m}\\ {\textbf{Zero Exponent Definition}}&{a^0=1, a \ne 0}\\ {\textbf{Quotient to a Power Property}}&{(\frac{a}{b})^m=\frac{a^m}{b^m}, b \ne 0}\\ \nonumber \end{array}\]
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_333%3A_Introduction_to_College_Algebra/02%3A_Equations_and_Inequalities/2.06%3A_Other_Types_of_EquationsRational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the v...Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1. Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping. We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index.
- https://math.libretexts.org/Courses/Las_Positas_College/Book%3A_College_Algebra/02%3A_Equations_and_Inequalities/2.06%3A_Other_Types_of_EquationsRational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the v...Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1. Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping. We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index.
- https://math.libretexts.org/Workbench/College_Algebra_2e_(OpenStax)/02%3A_Equations_and_Inequalities/2.07%3A_Other_Types_of_EquationsRational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the v...Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1. Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping. We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index.
- https://math.libretexts.org/Workbench/1250_Draft_3/01%3A_Prerequisites/1.04%3A_Radicals_and_Rational_ExpressionsJust as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplif...Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator.
- https://math.libretexts.org/Courses/Mission_College/Math_1X%3A_College_Algebra_w__Support_(Sklar)/08%3A_Support_Math_Topics/8.03%3A_Radicals_and_Rational_ExpressionsJust as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplif...Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator.
- https://math.libretexts.org/Courses/Western_Connecticut_State_University/Draft_Custom_Version_MAT_131_College_Algebra/02%3A_Equations_and_Inequalities/2.06%3A_Other_Types_of_EquationsRational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the v...Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1. Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping. We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index.
- https://math.libretexts.org/Courses/Chabot_College/Chabot_College_College_Algebra_for_BSTEM/02%3A_Equations_and_Inequalities/2.06%3A_Other_Types_of_EquationsRational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the v...Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1. Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping. We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index.
- https://math.libretexts.org/Courses/Palo_Alto_College/College_Algebra/01%3A_Prerequisites/1.04%3A_Radicals_and_Rational_ExpressionsJust as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplif...Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator.
- 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, \((a\cdot b)^3=a\cdot b \cdot a\cdot b \cdot a\cdot b=a\cdot a \cdot a\cdot b \cdot b\cdot b=a^3\cdot b^3\).
