# Extend the properties of exponents to rational exponents

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• Extend properties of integer exponents to rational exponents and write expressions with rational exponents as radicals.

• Solve real-world problems in which rational exponents arise.

Students have encountered square roots and cube roots in Grades 6–8. Now that they have the real numbers at their disposal they can contemplate more complicated numerical expressions involving radicals and fractional exponents. In this section they learn the rules for manipulating such expressions. They first review familiar exponent rules and remind themselves how they rewrite exponential expressions, particularly the rule $(x^a)^b = x^{ab}$. They investigate the consequences of extending this rule to rational exponents and see how it implies that $x^(a/b)=\sqrt[b]{x^a}$. The section continues with a modeling task using Kepler’s Law and then wraps up with a short reasoning task where students can work with rational exponents in decimal form.

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