Look at this expression.
[tex]$\frac{x^5 y^3 z^3}{x y^{-3} z^{-1}}$[/tex]
Which of the following is the simplest form of this expression?
A. [tex]$x^4 z^2$[/tex]
B. [tex]$x^4 y^6 z^4$[/tex]
C. [tex]$x^5 y^{-1} z^{-3}$[/tex]
D. [tex]$x^6 y^6 z^4$[/tex]
Answer (2)
Apply the quotient rule of exponents to each variable.
Simplify the exponents for each variable: x 5 − 1 = x 4 , y 3 − ( − 3 ) = y 6 , z 3 − ( − 1 ) = z 4 .
Combine the simplified terms: x 4 y 6 z 4 .
The simplest form of the expression is x 4 y 6 z 4 .
Explanation
Understanding the Expression We are given the expression x y − 3 z − 1 x 5 y 3 z 3 and we want to simplify it. We will use the quotient rule of exponents, which states that a n a m = a m − n .
Applying the Quotient Rule We apply the quotient rule to each variable separately:
For x : x x 5 = x 5 − 1 = x 4
For y : y − 3 y 3 = y 3 − ( − 3 ) = y 3 + 3 = y 6
For z : z − 1 z 3 = z 3 − ( − 1 ) = z 3 + 1 = z 4
Combining the Terms Now, we combine the simplified terms to get the simplest form of the expression:
x 4 y 6 z 4
Final Answer Therefore, the simplest form of the given expression is x 4 y 6 z 4 . Comparing this to the given options, we see that it matches option B.
Examples
Understanding how to simplify expressions with exponents is useful in many areas, such as calculating the volume of a rectangular prism or cylinder. For example, if the volume of a rectangular prism is given by V = l × w × h , and the length, width, and height are l = x 2 y , w = x y 3 , and h = x 3 y 2 , then the volume can be expressed as V = ( x 2 y ) × ( x y 3 ) × ( x 3 y 2 ) = x 6 y 6 . Simplifying expressions with exponents helps in determining the overall volume in terms of x and y .
The simplest form of the expression x y − 3 z − 1 x 5 y 3 z 3 is x 4 y 6 z 4 , which corresponds to option B. We used the quotient rule of exponents to arrive at this solution. Each variable was simplified separately before combining them together.
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