Questions: Use the relevant properties of exponents to simplify the following expression such that each base is only represented once and your answer only has positive exponents. x^4 y^7 z^0 / x^9 y^4

Solution Steps

To simplify the given expression, we will use the properties of exponents. Specifically, we will apply the quotient rule, which states that when dividing like bases, we subtract the exponents. We will also use the fact that any number raised to the power of zero is one, which allows us to eliminate \( z^0 \).

We start with the expression: \[ \frac{x^{4} y^{7} z^{0}}{x^{9} y^{4}} \]

Using the property that \( z^{0} = 1 \), we can simplify the expression to: \[ \frac{x^{4} y^{7}}{x^{9} y^{4}} \]

Now, we apply the quotient rule for exponents:

• For the base \( x \): \[ \frac{x^{4}}{x^{9}} = x^{4-9} = x^{-5} \]
• For the base \( y \): \[ \frac{y^{7}}{y^{4}} = y^{7-4} = y^{3} \]

Combining the results, we have: \[ \frac{y^{3}}{x^{5}} = y^{3} x^{-5} \]

To express the final result with only positive exponents, we rewrite \( x^{-5} \) as \( \frac{1}{x^{5}} \): \[ \frac{y^{3}}{x^{5}} \]

Final Answer