Simplify The Following Expressions:1. $22x^2 - 10x^2y + 8x^2y^2$2. $18r^2s^3 + 12r^3s^4$

In algebra, simplifying expressions is a crucial step in solving mathematical problems. It involves combining like terms and reducing the complexity of the expression. In this article, we will simplify two given algebraic expressions.

Expression 1: $22x^2 - 10x^2y + 8x^2y^2$

To simplify this expression, we need to identify the like terms. Like terms are the terms that have the same variable(s) with the same exponent(s). In this expression, the like terms are the terms with $x^2$ as the common factor.

22x^2 - 10x^2y + 8x^2y^2

We can factor out $x^2$ from each term:

x^2(22 - 10y + 8y^2)

Now, we can simplify the expression by combining the constants inside the parentheses:

x^2(22 - 10y + 8y^2) = x^2(22 + 8y^2 - 10y)

However, we can further simplify the expression by combining the constants:

x^2(22 + 8y^2 - 10y) = x^2(8y^2 - 10y + 22)

Expression 2: $18r^2s^3 + 12r^3s^4$

To simplify this expression, we need to identify the like terms. Like terms are the terms that have the same variable(s) with the same exponent(s). In this expression, the like terms are the terms with $r^2s^3$ as the common factor.

18r^2s^3 + 12r^3s^4

We can factor out $r^2s^3$ from the first term:

18r^2s^3 = 18r^2s^3(1)

However, we cannot factor out $r^2s^3$ from the second term because the exponent of $r$ is different. Therefore, we need to rewrite the expression as:

18r^2s^3 + 12r^3s^4 = 18r^2s^3 + 12r^3s^4

Now, we can simplify the expression by factoring out the greatest common factor (GCF) of the coefficients:

18r^2s^3 + 12r^3s^4 = 6(3r^2s^3 + 2r^3s^4)

However, we can further simplify the expression by factoring out the GCF of the variables:

6(3r^2s^3 + 2r^3s^4) = 6r^2s^3(3 + 2rs)

Conclusion

Simplifying algebraic expressions is an essential step in solving mathematical problems. By identifying like terms and combining them, we can reduce the complexity of the expression and make it easier to work with. In this article, we simplified two given algebraic expressions by factoring out common factors and combining like terms.

Tips and Tricks

• When simplifying algebraic expressions, always look for like terms and combine them.
• Factor out common factors, such as coefficients and variables, to simplify the expression.
• Use the distributive property to expand expressions and combine like terms.
• Simplify expressions by combining constants and variables.

Practice Problems

1. Simplify the expression: $15x^3y^2 - 20x^3y^2 + 25x^3y^2$
2. Simplify the expression: $24r^2s^3 + 36r^3s^4 - 48r^2s^3$
3. Simplify the expression: $18x^2y^3 - 12x^2y^3 + 20x^2y^3$

Answer Key

1. $20x^3y^2$
2. $24r^2s^3 + 36r^3s^4$
3. $26x^2y^3$

References

• "Algebra" by Michael Artin
• "Calculus" by Michael Spivak
• "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  Simplify the Given Algebraic Expressions: Q&A =====================================================

In our previous article, we simplified two given algebraic expressions. In this article, we will answer some frequently asked questions (FAQs) related to simplifying algebraic expressions.

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to identify the like terms. Like terms are the terms that have the same variable(s) with the same exponent(s).

Q: How do I identify like terms in an algebraic expression?

A: To identify like terms in an algebraic expression, you need to look for the terms that have the same variable(s) with the same exponent(s). For example, in the expression $22x^2 - 10x^2y + 8x^2y^2$, the like terms are the terms with $x^2$ as the common factor.

Q: What is the difference between a like term and a unlike term?

A: A like term is a term that has the same variable(s) with the same exponent(s). A unlike term is a term that has different variable(s) or different exponent(s). For example, in the expression $22x^2 - 10x^2y + 8x^2y^2$, the term $22x^2$ is a like term, while the term $8x^2y^2$ is a unlike term.

Q: How do I simplify an algebraic expression with multiple like terms?

A: To simplify an algebraic expression with multiple like terms, you need to combine the like terms by adding or subtracting their coefficients. For example, in the expression $22x^2 - 10x^2y + 8x^2y^2$, you can combine the like terms by adding their coefficients:

22x^2 - 10x^2y + 8x^2y^2 = x^2(22 - 10y + 8y^2)

Q: What is the distributive property in algebra?

A: The distributive property in algebra is a rule that allows you to multiply a single term by multiple terms. It states that for any numbers $a$, $b$, and $c$, the following equation holds:

a(b + c) = ab + ac

Q: How do I use the distributive property to simplify an algebraic expression?

A: To use the distributive property to simplify an algebraic expression, you need to multiply a single term by multiple terms. For example, in the expression $6(3x^2 + 2x^3)$, you can use the distributive property to simplify the expression:

6(3x^2 + 2x^3) = 18x^2 + 12x^3

Q: What is the greatest common factor (GCF) in algebra?

A: The greatest common factor (GCF) in algebra is the largest factor that divides all the terms in an expression. For example, in the expression $12x^2y^3 + 18x^2y^3$, the GCF is $6x^2y^3$.

Q: How do I find the GCF of multiple terms in an algebraic expression?

A: To find the GCF of multiple terms in an algebraic expression, you need to identify the largest factor that divides all the terms. For example, in the expression $12x^2y^3 + 18x^2y^3$, you can find the GCF by identifying the largest factor that divides both terms:

12x^2y^3 + 18x^2y^3 = 6x^2y^3(2 + 3)

Conclusion

Simplifying algebraic expressions is an essential step in solving mathematical problems. By identifying like terms, combining them, and using the distributive property, we can simplify complex expressions and make them easier to work with. In this article, we answered some frequently asked questions (FAQs) related to simplifying algebraic expressions.

Tips and Tricks

• Always identify like terms and combine them to simplify an algebraic expression.
• Use the distributive property to multiply a single term by multiple terms.
• Find the greatest common factor (GCF) of multiple terms to simplify an algebraic expression.
• Simplify expressions by combining constants and variables.

Practice Problems

1. Simplify the expression: $15x^3y^2 - 20x^3y^2 + 25x^3y^2$
2. Simplify the expression: $24r^2s^3 + 36r^3s^4 - 48r^2s^3$
3. Simplify the expression: $18x^2y^3 - 12x^2y^3 + 20x^2y^3$

Answer Key

1. $20x^3y^2$
2. $24r^2s^3 + 36r^3s^4$
3. $26x^2y^3$

References

• "Algebra" by Michael Artin
• "Calculus" by Michael Spivak
• "Mathematics for Computer Science" by Eric Lehman and Tom Leighton