Describe the steps for simplifying the expression 3a^3 + 4 - 4a - 2a^3 + 6 + 12a - b^3. Be specific and be sure to include the meaning of like terms in your explanation.
3a^3 + 4 - 4a - 2a^3 + 6 + 12a - b^3
Before we simplify this, I'll just write some terms:
*Variable: in this case, the variables are a and b
*Numerical coefficient: the number before the variables
*Exponent: number raised to certain term
*Constant: the numbers (w/o variables) in an expression
To simplify this, first we determine which terms have the same variable. Of course their exponents must be the same also. These terms can be combined.
Note that 3a^3 and -2a^3 have the same variable and its exponent. We combine their numerical coefficients, while the variable & exponent stays the same:
3a^3 - 2a^3 = (3-2) a^3 = 1a^3 = a^3
Usually, we don't write 1 as numerical coefficient, as it is already understood.
We also do this to -4a and 12a:
-4a + 12a = (-4+12) a = 8a
And to the constants 4 and 6:
4 + 6 = 10
Combining everything, we got
a^3 + 8a + 10 - b^3
Hope this helps~ :)
To simplify the expression 3a^3 + 4 - 4a - 2a^3 + 6 + 12a - b^3, we need to combine like terms. Like terms are terms that have the same variables raised to the same powers.
Let's break down the expression and identify the like terms:
3a^3 and -2a^3 are like terms because they both have the variable a raised to the power of 3.
4 and 6 are like terms because they are both constants.
-4a and 12a are like terms because they both have the variable a raised to the power of 1 (or simply a, since the exponent is 1).
-b^3 is a term on its own, as there are no other terms with b raised to the power of 3.
Now that we have identified the like terms, we can combine them:
Combining 3a^3 and -2a^3 gives us 3a^3 - 2a^3 = a^3.
Combining 4 and 6 gives us 4 + 6 = 10.
Combining -4a and 12a gives us -4a + 12a = 8a.
Our expression now becomes: a^3 + 10 + 8a - b^3.
So, simplifying the given expression involves combining the like terms and simplifying them to their simplest form.