## Equations With Rational Exponents Assignment Discovery

The goal of this unit is to help the students gain a conceptual understanding of rational exponents, relate them back to operations with rational numbers, and then using them in expressions. To begin, students make a list of their prior knowledge of exponents. I give them 2 minutes to make a list in pairs and then we compile a class list.

Students then chart and graph the first seven ordered pairs in the warm up scenario(**Math Practice 2**). The graph should be points without a curve running through them. I remind the students to label both the table and the axes on the graph. These will vary and this is a good opportunity for students to explain why they chose the labels they did (**Math Practice 3**). This is their introduction to exponential functions. We extend the graph to include the full exponential curve rather than just the integer pairs. By analyzing the curve as a class, the existence of rational exponents will become apparent.

Using their prior knowledge of multiplicative inverses and quadratic and cubic equations, the students have an opportunity to "discover" the meaning of the denominator in a rational exponent (**Math Practice 1**).

You multiply rational expressions in the same way as you multiply fractions of rational numbers. In other words you multiply the numerators with each other and the denominators with each other.

**Example**

$$\frac{4xy^{2}}{3y}\cdot \frac{2x}{4y}=$$

$$=\frac{4xy^{2}\cdot 2x}{3y\cdot 4y}=\frac{8x^{2}y^{2}}{12y^{2}}=\frac{{\color{red} {\not}}{4}\cdot 2x^{2}{\color{red} {\not}}{y^{2}}}{{\color{red}{ \not}}{4}\cdot 3{\color{red} {\not}}{y^{2}}}=\frac{2x^{2}}{3}$$

You can either start by multiplying the expressions and then simplify the expression as we did above or you could start by simplifying the expressions when it's still in fractions and then multiply the remaining terms e.g.

$$\frac{4xy^{2}}{3y}\cdot \frac{2x}{4y}=$$

$$=\frac{{\color{red} {\not}}{4}xy^{2}\cdot 2x}{3y\cdot{\color{red} {\not}}{4}y}=\frac{2x^{2}{\color{red} {\not}}{y^{2}}}{3{\color{red} {\not}}{y^{2}}}=\frac{2x^{2}}{3} $$

**Video lesson**

Multiply the rational expressions

$$\frac{4xy^{2}}{3y}\cdot \frac{5x^{2}}{2y}$$

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