Simplify the following: $\left(-6 y^4\right)^2\left(-2 y^6\right)^3$ =

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 .
Calculate ( − 6 ) 2 = 36 and ( − 2 ) 3 = − 8 .
Multiply the results and apply the product of powers rule: − 288 y 26 .
− 288 y 26 ​

Explanation

Initial Expression Let's simplify the given expression step by step: ( − 6 y 4 ) 2 ( − 2 y 6 ) 3

Applying Power of a Product Rule First, we apply the power of a product rule, which states that ( ab ) n = a n b n . So we have: ( − 6 ) 2 ( y 4 ) 2 ( − 2 ) 3 ( y 6 ) 3

Applying Power of a Power Rule Next, we apply the power of a power rule, which states that ( a m ) n = a mn . So we have: ( − 6 ) 2 y 4 × 2 ( − 2 ) 3 y 6 × 3 ( − 6 ) 2 y 8 ( − 2 ) 3 y 18

Calculating the Constants Now, we calculate ( − 6 ) 2 and ( − 2 ) 3 :
( − 6 ) 2 = ( − 6 ) × ( − 6 ) = 36 ( − 2 ) 3 = ( − 2 ) × ( − 2 ) × ( − 2 ) = − 8 So the expression becomes: 36 y 8 ( − 8 ) y 18

Multiplying and Simplifying Now, we multiply the constants and use the product of powers rule, which states that a m a n = a m + n :
36 × ( − 8 ) y 8 y 18 − 288 y 8 + 18 − 288 y 26

Final Result Therefore, the simplified expression is: − 288 y 26


Examples
Understanding how to simplify expressions with exponents is crucial in many areas, such as physics and engineering. For example, when calculating the energy of a photon, E = h f , where f is the frequency, and frequency might be expressed in terms of other variables with exponents. Simplifying such expressions allows for easier calculations and a better understanding of the relationships between different physical quantities. Similarly, in computer graphics, transformations like scaling involve raising coordinates to certain powers, and simplifying these expressions is essential for efficient rendering.

Answered by GinnyAnswer | 2025-07-08

To simplify ( − 6 y 4 ) 2 ( − 2 y 6 ) 3 , we use exponent rules to get the answer − 288 y 26 . We calculated powers, combined terms, and applied the properties of exponents step-by-step. The final result is − 288 y 26 .
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Answered by Anonymous | 2025-07-27