Which expression is equal to 4mn828m5n3 – In the realm of algebra, the expression 4mn828m5n3 stands as an intriguing mathematical enigma. This intricate combination of variables and numerical coefficients presents a challenge to simplify and unravel its true identity. Embarking on this journey, we delve into the fascinating world of algebraic expressions, uncovering the secrets that lie within 4mn828m5n3.
Through a meticulous examination of its structure and application of fundamental algebraic principles, we will systematically dissect this expression, revealing its underlying simplicity and elegance. Along the way, we will encounter the power of factoring, the distributive property, and the art of combining like terms, all of which serve as essential tools in our quest to conquer this algebraic puzzle.
Algebraic Expressions
Algebraic expressions are mathematical expressions that contain variables, constants, and operations. They can be used to represent a wide variety of mathematical concepts, including equations, inequalities, and functions.
One common type of algebraic expression is the monomial, which is a single term consisting of a coefficient and a variable raised to a power. For example, the expression 4mn is a monomial with a coefficient of 4 and variables m and n.
Monomials can be combined using the operations of addition, subtraction, and multiplication to form more complex algebraic expressions. For example, the expression 4mn + 828m5n3 is an algebraic expression that consists of two monomials.
Table of Equivalent Expressions
The following table shows three different algebraic expressions that are all equal to 4mn828m5n3:
| Expression | Value | Explanation |
|---|---|---|
| 4mn828m5n3 | 4mn828m5n3 | The original expression |
| 4m(207m4n3) | 4mn828m5n3 | Factoring out a common factor of 4m |
| 4(mn207m4n3) | 4mn828m5n3 | Distributing the coefficient 4 over the parentheses |
Factoring and Expanding
Factoring is the process of breaking down an algebraic expression into smaller factors. Expanding is the reverse process of combining factors to form a larger expression.
Steps for Factoring 4mn828m5n3
- Factor out the greatest common factor (GCF) of 4m.
- Factor the remaining expression, 207m4n3, as a product of two binomials.
- Combine the two factors to get the final factored expression.
Example of Expanding a Factored Expression, Which expression is equal to 4mn828m5n3
The expression (2x + 3)(x – 5) can be expanded using the distributive property:
(2x + 3)(x - 5) = 2x(x - 5) + 3(x - 5) = 2x^2 - 10x + 3x - 15 = 2x^2 - 7x - 15
Distributive Property: Which Expression Is Equal To 4mn828m5n3
The distributive property is a mathematical property that states that the product of a sum and a factor is equal to the sum of the products of the factor and each term in the sum.
Using the Distributive Property to Simplify 4mn828m5n3
The expression 4mn828m5n3 can be simplified using the distributive property:
4mn828m5n3 = 4(mn + 828m5n3) = 4mn + 4(828m5n3) = 4mn + 3312m5n3
Example of Using the Distributive Property
The expression 3(x + 2) can be simplified using the distributive property:
3(x + 2) = 3x + 3(2) = 3x + 6
Combining Like Terms
Combining like terms is the process of adding or subtracting terms that have the same variable and exponent.
Table of Simplified Expressions
The following table shows three different algebraic expressions that can be simplified by combining like terms:
| Original Expression | Simplified Expression |
|---|---|
| 4mn + 828m5n3 | 3312m5n3 + 4mn |
| 3x + 2x + 5 | 5x + 5 |
4y
|
2y + 7 |
FAQ Explained
What is the significance of 4mn828m5n3 in algebra?
4mn828m5n3 represents a complex algebraic expression that provides a unique opportunity to apply and reinforce fundamental algebraic principles, including factoring, the distributive property, and combining like terms.
How can I approach simplifying 4mn828m5n3 effectively?
To simplify 4mn828m5n3 effectively, it is crucial to break down the expression into manageable components. Start by identifying any common factors or terms that can be combined. Utilize the distributive property to expand and simplify the expression further. Finally, combine like terms to obtain the simplest possible form.
What are the key takeaways from simplifying 4mn828m5n3?
Simplifying 4mn828m5n3 reinforces the importance of understanding algebraic operations and their application in simplifying complex expressions. It highlights the power of factoring, the distributive property, and combining like terms as essential techniques for algebraic simplification.
