Polynomials-Target+A-Simplify+Exponents-Guided+Learning

Polynomials Target A: Simplify (Write equivalent) exponential expressions involving products and quotients

 * [[image:Discuss this.png width="208" height="117"]] || Looking at the expression (2 3 )(2 2 ), Kevin thinks that the expession is equivalent to 2 6 , Carol thinks that it is equivalent to 2 5 , and Charlie thinks that it is equivalent to 4 6 . Whom do you agree with and why? ||

Before we begin to simplify an exponential expression involving products, let's recall what it means to write an exponential expression in expanded form. For example:
 * Using Property of Products**

Now let's look at an example of multiplying single variable expressions with exponents. We will work with expanding the expressions to see what happens when we multiply. As we can see that when we expand both expressions in the first example we end up with "5" x's multiplied together, then using exponents we can write an equivalent expression of x 5. We must know that we can only do this if the bases of each exponential are the same, that is they are both x's or both y's, etc.

What I would like for you to notice is there is a pattern when multiplying, a more efficient way to mulitply like bases instead of expanding the exponential each time, which could be cumbersome if we had an expression like x 25. Let's look at our examples again, but let's focus on the original two expressions' exponents and the final product. What do you notice? Hopefully this is a "AHA" moment! Yes! It is true when the expressions have like bases and you are finding the product you add the exponents!! media type="custom" key="27722975"

Now let's do some different examples that are a bit more involved.

Before we begin simplifying exponential expressions involving quotients, let's first investigate what happens when we divide a single variable expression with exponents. Start by expanding the numerator and the denominator. Then, divide out each "pair" of the same variable (we know that if we divide something by itself, it equals 1) to see what is left.
 * Using Property of Quotients**

Is there a more efficient way to divide out exponents? It would take a while to always expand the power. Using the same examples as above, is there a noticeable pattern that would get us from the original expression to the simplified expression? How do we get from 5 and 3 to 2? How do we get from 6 and 3 to 3? What about from 4 and 1 to 3? We use subtraction! So the quotient rule when simplifying exponential expressions is to subtract the exponents (always subtract the exponent in the denominator from the exponent in the numerator) of the powers with the SAME base. Let's move on to some expressions where the numerators and denominators have multiple variables and coefficients.

When simplifying an expression using the property of quotients, sometimes it is helpful to think about the fraction (or division) as having different components. Work through the division of the coefficients first, and then each variable separately. Let's take a look at the examples below.


 * Polynomials-Target A-Simplify Exponents-Quick Check:**

Simplify the expressions.

math 1) \space\large5x^{2}y\cdot 3x^{7}y^{^{4}}\ math

math 2) \space\LARGE\frac{15a^{6}b^{3}}{5a^{5}b}\ math

math 3) \space\LARGE\frac{24g^{5}h^{7}}{16g^{3}h^{3}}\ math

Polynomials-Target A-Simplify Exponents-Quick Check Solutions