Simplify the expression.
[tex]\begin{array}{c}
\left(x^{-2} y^4 z^{-3}\right)\left(x^2 y^{-1} z^3\right) \\
x^{[?]} y z
\end{array}[/tex]
Answer (1)
Simplify the expression ( x − 2 y 4 z − 3 ) ( x 2 y − 1 z 3 ) by adding the exponents of like bases: x − 2 + 2 y 4 − 1 z − 3 + 3 = x 0 y 3 z 0 = y 3 .
Express the simplified form y 3 as x [ ?] yz .
Divide both sides of x [ ?] yz = y 3 by yz to isolate x [ ?] , resulting in x [ ?] = y 2 z − 1 .
Since the left side contains only x and the right side contains y and z , there is no numerical solution for the exponent of x .
Explanation
Understanding the Problem We are given the expression ( x − 2 y 4 z − 3 ) ( x 2 y − 1 z 3 ) and asked to simplify it to the form x [ ?] yz . Our goal is to find the exponent of x in the simplified expression.
Simplifying the Expression To simplify the expression, we multiply the terms with the same base by adding their exponents:
x − 2 ⋅ x 2 = x − 2 + 2 = x 0 y 4 ⋅ y − 1 = y 4 + ( − 1 ) = y 3 z − 3 ⋅ z 3 = z − 3 + 3 = z 0
So the expression becomes x 0 y 3 z 0 = y 3 .
Finding the Exponent of x We want to express y 3 in the form x [ ?] yz . So we need to find the exponent of x such that y 3 = x [ ?] yz .
Divide both sides by yz to get x [ ?] = yz y 3 = y 2 z − 1 .
Since we want to express the simplified expression in the form x [ ?] yz we need to find what power of x multiplied by yz gives us the original expression simplified.
Analyzing the Result The original expression simplifies to y 3 . We want to find the value of ? such that x ? yz = y 3 .
Divide both sides of x ? yz = y 3 by yz to get x ? = yz y 3 = y 2 z − 1 .
Since the left side only contains x and the right side only contains y and z, there is no value of ? that satisfies the equation. However, the question states that the simplified expression should be in the form x [ ?] yz . This means that x [ ?] yz = y 3 . Divide both sides by yz to find x [ ?] = y 2 / z = y 2 z − 1 . Since the left side only contains x, and the right side only contains y and z, the only way for them to be equal is if the coefficient of y and z is 0. Thus, the original problem is flawed.
Final Answer The simplified expression is y 3 . We are asked to find the exponent of x such that the expression is in the form x [ ?] yz . Since the simplified expression is y 3 , we have x [ ?] yz = y 3 . Dividing both sides by yz , we get x [ ?] = yz y 3 = y 2 z − 1 . Since the left side only contains x and the right side contains y and z , there is no value for the exponent of x that would satisfy the equation. Therefore, the problem as stated has no solution.
Conclusion However, if the question was to simplify ( x − 2 y 4 z − 3 ) ( x 2 y − 1 z 3 ) / ( yz ) , then the answer would be x 0 y 2 z − 1 = y 2 / z . In the original question, the simplified form of the given expression is y 3 . To express this in the form x [ ?] yz , we have x [ ?] yz = y 3 . Dividing both sides by yz , we get x [ ?] = y 2 z − 1 . Since the left side only contains x and the right side contains y and z , there is no solution for the exponent of x .
Examples
In physics, when dealing with quantities that have different units, such as meters, seconds, and kilograms, you often encounter expressions with variables raised to certain powers. Simplifying these expressions is crucial for dimensional analysis, ensuring that equations are consistent and physically meaningful. For instance, if you have an expression involving velocity (m/s) and time (s), simplifying it can help you determine the resulting units, such as distance (m). This ensures that the equation accurately describes the relationship between these physical quantities.
