Choose the correct simplification of $\frac{x^{12} z^{11}}{x^2 z^4}$.
A. $x^{10} z^7$
B. $x^{14} z^{15}$
C. $\frac{1}{x^{10} z^7}$
D. $\frac{1}{x^{14} z^{15}}$
Answer (1)
Apply the quotient rule of exponents: a n a m = a m − n .
Simplify the x terms: x 2 x 12 = x 10 .
Simplify the z terms: z 4 z 11 = z 7 .
Combine the simplified terms: x 10 z 7 .
Explanation
Understanding the Problem We are given the expression x 2 z 4 x 12 z 11 to simplify. We need to choose the correct simplified expression from the given options. The properties of exponents will be used to simplify the expression.
Applying the Quotient Rule To simplify the expression, we will use the quotient rule of exponents, which states that when dividing terms with the same base, we subtract the exponents: a n a m = a m − n .
Simplifying x Terms First, let's simplify the x terms: x 2 x 12 = x 12 − 2 = x 10 .
Simplifying z Terms Next, let's simplify the z terms: z 4 z 11 = z 11 − 4 = z 7 .
Combining Simplified Terms Now, we combine the simplified x and z terms to get the final simplified expression: x 10 z 7 .
Final Answer Therefore, the correct simplification of x 2 z 4 x 12 z 11 is x 10 z 7 .
Examples
Understanding how to simplify expressions with exponents is useful in many areas, such as calculating areas and volumes in geometry, or in physics when dealing with units and measurements. For example, if you are calculating the volume of a cylinder with radius x 5 and height z 3 , the area of the base would be π ( x 5 ) 2 = π x 10 and the volume would be π x 10 z 3 . Simplifying expressions helps in making these calculations easier and more manageable.
