What is the only rule that could NOT apply when simplifying $\left(-1 x z^2\right)^3\left(x^3\right)^2$?
A. negative rule
B. product of a power rule
C. power of a power rule
D. product rule
Answer (1)
Apply the power of a product rule: ( − 1 x z 2 ) 3 = ( − 1 ) 3 x 3 ( z 2 ) 3 .
Use the power of a power rule: ( z 2 ) 3 = z 6 and ( x 3 ) 2 = x 6 .
Apply the product rule: x 3 x 6 = x 9 .
Identify that the product of a power rule is not applicable in this simplification.
The only rule that could NOT apply is product of a power rule .
Explanation
Understanding the Problem We are given the expression ( − 1 x z 2 ) 3 ( x 3 ) 2 and asked to identify which of the given rules (negative rule, product of a power rule, power of a power rule, product rule) could NOT apply when simplifying it.
Applying Power of a Product Rule First, apply the power of a product rule to the first term: ( − 1 x z 2 ) 3 = ( − 1 ) 3 x 3 ( z 2 ) 3 .
Simplifying with Power of a Power Rule Next, apply the power of a power rule to simplify ( z 2 ) 3 = z 2 × 3 = z 6 . Also, ( − 1 ) 3 = − 1 . Thus, ( − 1 x z 2 ) 3 = − 1 x 3 z 6 = − x 3 z 6 .
Applying Power of a Power Rule Again Apply the power of a power rule to the second term: ( x 3 ) 2 = x 3 × 2 = x 6 .
Multiplying Terms Multiply the simplified terms: ( − x 3 z 6 ) ( x 6 ) = − x 3 x 6 z 6 .
Using the Product Rule Apply the product rule (product of powers rule): x 3 x 6 = x 3 + 6 = x 9 .
Identifying the Non-Applicable Rule The final simplified expression is − x 9 z 6 .
Now consider each rule: The negative rule was used to evaluate ( − 1 ) 3 = − 1 . The power of a power rule was used to evaluate ( z 2 ) 3 = z 6 and ( x 3 ) 2 = x 6 . The product rule was used to evaluate x 3 x 6 = x 9 .
The only rule that could NOT apply is the product of a power rule, because we are not multiplying terms with the same base raised to different powers before applying other rules. We are multiplying powers of powers.
Final Answer Therefore, the only rule that could NOT apply when simplifying the given expression is the product of a power rule.
Examples
When simplifying expressions in physics, such as calculating the volume of a complex shape or determining the energy of a system, understanding exponent rules is crucial. For example, if you're calculating the kinetic energy of an object with a changing velocity, you might encounter terms with exponents that need simplification. Knowing which rules apply and which don't ensures accurate and efficient calculations, preventing errors in your analysis.
