Simplify the expression. Evaluate coefficients if the exponent is an integer.
[tex]\left(\frac{x^{-1} y}{z^4}\right)^{-4}[/tex]
Answer (2)
Apply the power of a quotient rule: ( b a ) n = b n a n .
Apply the power of a product rule: ( ab ) n = a n b n .
Apply the power of a power rule: ( a m ) n = a mn .
Rewrite the expression with positive exponents: y 4 x 4 z 16 .
Explanation
Understanding the Problem We are asked to simplify the expression \left(\frac{x^{-1} y}{z^4}\\right)^{-4} . This involves using exponent rules to simplify the expression and rewrite it with positive exponents.
Applying the Power of a Quotient Rule First, we apply the power of a quotient rule, which states that ( b a ) n = b n a n . Applying this rule, we get: ( z 4 x − 1 y ) − 4 = ( z 4 ) − 4 ( x − 1 y ) − 4
Applying the Power of a Product Rule Next, we apply the power of a product rule, which states that ( ab ) n = a n b n . Applying this rule to the numerator, we get: ( z 4 ) − 4 ( x − 1 y ) − 4 = ( z 4 ) − 4 ( x − 1 ) − 4 y − 4
Applying the Power of a Power Rule Now, we apply the power of a power rule, which states that ( a m ) n = a mn . Applying this rule to each term, we get: ( z 4 ) − 4 ( x − 1 ) − 4 y − 4 = z ( 4 ) ( − 4 ) x ( − 1 ) ( − 4 ) y − 4 = z − 16 x 4 y − 4
Rewriting with Positive Exponents Finally, we rewrite the expression with positive exponents. Recall that a − n = a n 1 . Therefore, y − 4 = y 4 1 and z − 16 = z 16 1 , so z − 16 1 = z 16 . Thus, we have: z − 16 x 4 y − 4 = y 4 x 4 ⋅ z 16 = y 4 x 4 z 16
Final Answer Therefore, the simplified expression is y 4 x 4 z 16 .
Examples
Understanding exponent rules is crucial in many fields, such as physics and engineering, where you often deal with very large or very small numbers. For example, when calculating the force of gravity between two objects, you use the inverse square law, which involves negative exponents. Simplifying expressions with exponents allows engineers to accurately calculate forces, distances, and other physical quantities.
To simplify the expression ( z 4 x − 1 y ) − 4 , we applied exponent rules systematically. After rewriting the expression with positive exponents, we found that it simplifies to y 4 x 4 z 16 . This process involves using the power of a quotient, product, and power of a power rules.
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