To simplify the expression \( 3x^3y^{-4} - 12x^7y^4 \) using the properties of integer exponents, we need to combine the terms properly.
First, we can factor out the common terms from both monomials. The common factors will be \( 3x^3 \) because it has the lowest power of \( x \) and we can also consider the lowest power of \( y \), which is \( y^{-4} \).
Let's rewrite both terms by factoring out \( 3x^3y^{-4} \):
- The first term \( 3x^3y^{-4} \) stays as it is.
- The second term \( 12x^7y^4 \) can be written as: \[ 12x^7y^4 = 12x^3x^4y^{-4}y^8 = 12x^3y^{-4}(x^4y^8) \]
Now, factoring out \( 3x^3y^{-4} \): \[ = 3x^3y^{-4}(1 - 4x^4y^8) \]
Now we can look for equivalent algebraic expressions to represent this. The options provided do not directly correspond with the factorization, so we should expand this reasoning further.
Simplify this inside the parentheses: If we calculate \( 3(4) \) for the coefficients: \[ 12x^3y^{-4} = 4 \cdot 3x^3y^{-4} \Rightarrow 3x^3y^{-4}(1 - 4x^4y^8) \]
It might not seem straightforward from the response choices available. We can combine terms without involving negative exponents:
Rearranging \[ = 3x^3y^{-4}(1 - 4x^4y^8) = 3 \cdot 4x^3y^{-4} \]
In looking over the expression choices provided, none simplify exactly to the apparent forms. None of the options are simple representations of this polynomial.
Thus, the appropriate final answer simplistically by removing negative exponents can also be as \( 14x^4y^8 \) upon multiplying.
Final check: None of the options fit perfectly, as results must be added correctly from above, but the breakdown shows 12 from \( -12 \) was missed to give it context. Thus, The best of these responses by discussing gives \( 14x^4y^8 \) as a matching form.
Thus, the response is: 14x^4y^8.
5 answers