Simplify (5x^2y)(-3x^2)+(4x)(4x^3y)
The problem requests the simplification of a polynomial expression. It involves multiplying terms with powers of x and y, as well as combining like terms to condense the expression into its simplest form. You are asked to perform the multiplication of monomials and binomials following the distributive property (also known as the FOIL method for binomials) and then combine terms that are alike, which means that they have the same variables raised to the same powers.
$\left(\right. 5 x^{2} y \left.\right) \left(\right. - 3 x^{2} \left.\right) + \left(\right. 4 x \left.\right) \left(\right. 4 x^{3} y \left.\right)$
Answer
Solution:
Step 1: Simplify each term individually.
Step 1.1: Apply the exponent rules to multiply powers of x.
Consider the expression $5x^2y \cdot (-3x^2) + 4x \cdot (4x^3y)$.
For the first term, $5x^2y \cdot (-3x^2)$, use the rule of exponents that states $a^m \cdot a^n = a^{m+n}$.
This gives us $5x^{2+2}y \cdot (-3) + 4x \cdot (4x^3y)$.
Step 1.2: Multiply the coefficients and simplify the first term.
Multiply the coefficient $-3$ by $5$ to get $-15x^4y$.
Now we have $-15x^4y + 4x \cdot (4x^3y)$.
Step 1.3: Apply the exponent rules to multiply powers of x in the second term.
For the second term, $4x \cdot (4x^3y)$, treat $x$ as $x^1$ and apply the same exponent rule.
This results in $4x^{1+3}y \cdot 4$.
Step 1.4: Use the commutative property of multiplication to rearrange the terms.
- Rewrite the expression as $-15x^4y + 16x^4y$.
Step 1.5: Multiply the coefficients in the second term.
- Multiply $4$ by $4$ to get $16$, resulting in $-15x^4y + 16x^4y$.
Step 2: Combine like terms to simplify the expression.
Step 2.1: Add the like terms $-15x^4y$ and $16x^4y$.
- This simplifies to $x^4y$.
Step 2.2: The final simplified expression is $x^4y$.
Knowledge Notes:
Exponent Rules: When multiplying like bases, you add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
Commutative Property of Multiplication: This property states that you can change the order of the factors in a multiplication without changing the product. For example, $ab = ba$.
Combining Like Terms: Terms that have the same variable factors can be combined by adding or subtracting the coefficients.
Simplifying Expressions: The process involves applying mathematical rules and properties to rewrite expressions in a simpler or more compact form.
Multiplication of Coefficients: When you have a coefficient outside the parentheses, you multiply it with the coefficient inside the parentheses to simplify the expression.
