3.jpeg "(4s5t−7−2s−2t4)3")
Simplifying the Expression (4s^5t - 7 - 2s - 2t^4)^3
This expression involves a combination of variables, constants, and exponents. To simplify it, we'll use the following steps:
1. Understanding the Expression:
- (4s^5t - 7 - 2s - 2t^4)^3: This represents the entire expression in parentheses, cubed. Cubing means multiplying the expression by itself three times.
2. Expanding the Expression:
Instead of directly cubing the entire expression, it's easier to break it down. We can use the following formula: (a + b + c)^3 = a^3 + b^3 + c^3 + 3a^2b + 3a^2c + 3ab^2 + 3ac^2 + 6abc
Applying this to our expression, we get: (4s^5t)^3 + (-7)^3 + (-2s)^3 + 3(4s^5t)^2*(-7) + 3*(4s^5t)^2*(-2s) + 3*(4s^5t)(-7)^2 + 3(4s^5t)(-2s)^2 + 6(4s^5t)(-7)(-2s)*
3. Simplifying the Terms:
Now, let's simplify each term:
- (4s^5t)^3 = 64s^15t^3
- (-7)^3 = -343
- (-2s)^3 = -8s^3
- 3(4s^5t)^2(-7) = -336s^10t^2**
- 3(4s^5t)^2(-2s) = -96s^11t^2**
- 3(4s^5t)*(-7)^2 = 588s^5t*
- 3(4s^5t)*(-2s)^2 = 48s^7t*
- 6(4s^5t)(-7)(-2s) = 336s^6t*
4. Combining the Terms:
Finally, we combine all the simplified terms:
64s^15t^3 - 343 - 8s^3 - 336s^10t^2 - 96s^11t^2 + 588s^5t + 48s^7t + 336s^6t
5. Rearranging for Order:
It's conventional to arrange terms by decreasing order of exponents, and then alphabetically:
64s^15t^3 - 96s^11t^2 - 336s^10t^2 + 336s^6t + 48s^7t + 588s^5t - 8s^3 - 343
Conclusion:
The simplified form of the expression (4s^5t - 7 - 2s - 2t^4)^3 is 64s^15t^3 - 96s^11t^2 - 336s^10t^2 + 336s^6t + 48s^7t + 588s^5t - 8s^3 - 343. This process involved understanding the expression, expanding using a formula, simplifying each term, and finally combining and arranging the terms in a standard format.