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A4 Complex Numbers and Rational Exponents
In an earlier grade, students learned various techniques for solving quadratic equations including: solving by inspection (e.g. solving by knowing that -7 and 7 are the two numbers that square to make 49); taking square roots; graphing; completing the square; quadratic formula; and factoring. Students have also worked with radicals
including and in various geometric contexts and worked with expressions involving integer exponents to establish exponent rules.
In this unit, students: use what they know about exponents and radicals to extend
exponent rules to include rational exponents (e.g. ); solve various equations involving squares and square roots; develop the concept of complex numbers by deSning a new number whose square is -1; and using complex numbers to Snd solutions to quadratic equations.
The Srst set of lessons in the unit provides an opportunity for students to review what they know about exponent rules and radicals, and extend those patterns to make sense of
expressions with rational exponents. Students eventually add the rule , where and are whole numbers, to all the other exponent rules they know.
In the next set of lessons, students connect the and symbols with solutions to
quadratic and cubic equations. Students learn that a number is a square root of if it squares to make . In other words, square roots of are solutions to the equation . Students use the graph of to see that all positive numbers have two square roots, one positive and one negative. They learn the convention that the positive square root
is given the symbol , so the positive square root of is written and the negative
square root is written . Similarly, students use the graph of to see that all
numbers, positive or negative, have a single cube root, and so the solution to the equation is written as . Students solve equations like to Snd they have
one solution if is positive and no solutions if is negative because of the deSnition of the symbol (a positive number cannot be equal to a negative number). Critically,
students use this connection between square roots and solutions to to understand that squaring each side of an equation can sometimes introduce new solutions that aren't solutions to the original equation, and that applying the symbol on each of an equation
ignores the existence of negative square roots. This is important in later lessons when students must take care to account for both positive and negative square roots in the process of solving quadratic equations. Note that there are claims in these lessons like, “All numbers have exactly one cube root,” which would be more precisely stated as “All
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Course Guide Algebra