Simplify $(5 x^4 y^4 z)^3(x^4 y)^4$

Apply the power of a product rule to each term.
Apply the power of a power rule to simplify exponents.
Multiply the simplified terms together.
Apply the product of powers rule to combine like variables, resulting in the final simplified expression: 125 x 28 y 16 z 3 ​ .

Explanation

Understanding the Problem We are asked to simplify the expression ( 5 x 4 y 4 z ) 3 ( x 4 y ) 4 . To do this, we will use the properties of exponents.

Applying Power of a Product Rule First, we apply the power of a product rule to the first term: ( 5 x 4 y 4 z ) 3 = 5 3 ( x 4 ) 3 ( y 4 ) 3 z 3

Applying Power of a Power Rule Next, we apply the power of a power rule: ( x 4 ) 3 = x 4 × 3 = x 12 ( y 4 ) 3 = y 4 × 3 = y 12

Simplifying the First Term Now we simplify the first term: 5 3 x 12 y 12 z 3 = 125 x 12 y 12 z 3

Applying Power of a Product Rule Next, we apply the power of a product rule to the second term: ( x 4 y ) 4 = ( x 4 ) 4 y 4

Applying Power of a Power Rule Then, we apply the power of a power rule: ( x 4 ) 4 = x 4 × 4 = x 16

Simplifying the Second Term Now we simplify the second term: x 16 y 4

Multiplying the Simplified Terms Next, we multiply the simplified terms: ( 125 x 12 y 12 z 3 ) ( x 16 y 4 )

Applying the Product of Powers Rule Now, we apply the product of powers rule: x 12 x 16 = x 12 + 16 = x 28 y 12 y 4 = y 12 + 4 = y 16

Writing the Final Simplified Expression Finally, we write the simplified expression: 125 x 28 y 16 z 3


Examples
Understanding how to simplify expressions with exponents is crucial in many fields, such as physics and engineering. For example, when calculating the volume of a complex shape or determining the energy of a system, you often encounter expressions with exponents. Simplifying these expressions allows for easier calculations and a better understanding of the relationships between different variables. In computer graphics, transformations like scaling involve raising coordinates to certain powers, and simplifying these expressions can optimize rendering performance. This skill is also essential in financial mathematics when dealing with compound interest and exponential growth models.

Answered by GinnyAnswer | 2025-07-03

To simplify the expression ( 5 x 4 y 4 z ) 3 ( x 4 y ) 4 , we apply the power of a product and power of a power rules. The result simplifies to 125 x 28 y 16 z 3 . This shows how to combine exponents step-by-step effectively.
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Answered by Anonymous | 2025-07-04