Simplify The Expression: ( 2 T + 3 U ) ( 2 T − 3 U (2t + 3u)(2t - 3u ( 2 T + 3 U ) ( 2 T − 3 U ]

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements. One of the most common techniques used to simplify expressions is the distributive property, which states that for any real numbers a, b, and c, a(b + c) = ab + ac. In this article, we will use the distributive property to simplify the expression $(2t + 3u)(2t - 3u)$.

The Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside the parentheses. The distributive property can be written as:

a(b + c) = ab + ac

where a, b, and c are real numbers.

Simplifying the Expression

To simplify the expression $(2t + 3u)(2t - 3u)$, we will use the distributive property. We will multiply each term inside the first parentheses with each term inside the second parentheses.

$(2t + 3u)(2t - 3u)$

Using the distributive property, we can rewrite the expression as:

$(2t + 3u)(2t - 3u) = 2t(2t) + 2t(-3u) + 3u(2t) + 3u(-3u)$

Now, we can simplify each term:

$2t(2t) = 4t^2$

$2t(-3u) = -6tu$

$3u(2t) = 6tu$

$3u(-3u) = -9u^2$

Now, we can combine like terms:

$4t^2 - 6tu + 6tu - 9u^2$

The terms $-6tu$ and $6tu$ cancel each other out, so the expression simplifies to:

$4t^2 - 9u^2$

Conclusion

In this article, we used the distributive property to simplify the expression $(2t + 3u)(2t - 3u)$. We multiplied each term inside the first parentheses with each term inside the second parentheses and then combined like terms. The simplified expression is $4t^2 - 9u^2$. This technique is essential in algebra and is used to solve equations and manipulate mathematical statements.

Example Problems

1. Simplify the expression $(3x + 2y)(2x - 3y)$.
2. Simplify the expression $(4a - 2b)(3a + 2b)$.
3. Simplify the expression $(2c + 3d)(c - 2d)$.

Step-by-Step Solutions

1. Simplify the expression $(3x + 2y)(2x - 3y)$.

Using the distributive property, we can rewrite the expression as:

$(3x + 2y)(2x - 3y) = 3x(2x) + 3x(-3y) + 2y(2x) + 2y(-3y)$

Now, we can simplify each term:

$3x(2x) = 6x^2$

$3x(-3y) = -9xy$

$2y(2x) = 4xy$

$2y(-3y) = -6y^2$

Now, we can combine like terms:

$6x^2 - 9xy + 4xy - 6y^2$

The terms $-9xy$ and $4xy$ cancel each other out, so the expression simplifies to:

$6x^2 - 6y^2$

2. Simplify the expression $(4a - 2b)(3a + 2b)$.

Using the distributive property, we can rewrite the expression as:

$(4a - 2b)(3a + 2b) = 4a(3a) + 4a(2b) - 2b(3a) - 2b(2b)$

Now, we can simplify each term:

$4a(3a) = 12a^2$

$4a(2b) = 8ab$

$-2b(3a) = -6ab$

$-2b(2b) = -4b^2$

Now, we can combine like terms:

$12a^2 + 8ab - 6ab - 4b^2$

The terms $8ab$ and $-6ab$ cancel each other out, so the expression simplifies to:

$12a^2 - 4b^2$

3. Simplify the expression $(2c + 3d)(c - 2d)$.

Using the distributive property, we can rewrite the expression as:

$(2c + 3d)(c - 2d) = 2c(c) + 2c(-2d) + 3d(c) + 3d(-2d)$

Now, we can simplify each term:

$2c(c) = 2c^2$

$2c(-2d) = -4cd$

$3d(c) = 3cd$

$3d(-2d) = -6d^2$

Now, we can combine like terms:

$2c^2 - 4cd + 3cd - 6d^2$

The terms $-4cd$ and $3cd$ cancel each other out, so the expression simplifies to:

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Introduction

In our previous article, we used the distributive property to simplify the expression $(2t + 3u)(2t - 3u)$. We multiplied each term inside the first parentheses with each term inside the second parentheses and then combined like terms. The simplified expression is $4t^2 - 9u^2$. In this article, we will answer some frequently asked questions about simplifying expressions.

Q&A

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside the parentheses. The distributive property can be written as:

a(b + c) = ab + ac

where a, b, and c are real numbers.

Q: How do I use the distributive property to simplify expressions?

A: To use the distributive property to simplify expressions, you need to multiply each term inside the first parentheses with each term inside the second parentheses. Then, combine like terms.

Q: What are like terms?

A: Like terms are terms that have the same variable(s) raised to the same power. For example, $2x^2$ and $4x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have the expression $2x^2 + 4x^2$, you can combine the like terms by adding the coefficients: $2 + 4 = 6$, so the expression simplifies to $6x^2$.

Q: What if I have a negative sign in front of one of the terms?

A: If you have a negative sign in front of one of the terms, you need to distribute the negative sign to the term inside the parentheses. For example, if you have the expression $-(2t + 3u)$, you can distribute the negative sign by multiplying each term inside the parentheses with the negative sign: $-2t - 3u$.

Q: Can I use the distributive property to simplify expressions with more than two parentheses?

A: Yes, you can use the distributive property to simplify expressions with more than two parentheses. You need to multiply each term inside the first parentheses with each term inside the second parentheses, and then multiply each term inside the resulting expression with each term inside the third parentheses, and so on.

Q: What if I have a fraction in the expression?

A: If you have a fraction in the expression, you need to multiply each term inside the first parentheses with each term inside the second parentheses, and then multiply the resulting expression by the fraction. For example, if you have the expression $(2t + 3u)\frac{1}{2}$, you can multiply each term inside the parentheses with the fraction: $\frac{1}{2}(2t + 3u) = t + \frac{3}{2}u$.

Q: Can I use the distributive property to simplify expressions with exponents?

A: Yes, you can use the distributive property to simplify expressions with exponents. You need to multiply each term inside the first parentheses with each term inside the second parentheses, and then multiply the resulting expression by the exponent. For example, if you have the expression $(2t + 3u)^2$, you can multiply each term inside the parentheses with each term inside the other parentheses: $(2t + 3u)(2t + 3u) = 4t^2 + 6tu + 6tu + 9u^2$.

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions using the distributive property. We covered topics such as like terms, combining like terms, and using the distributive property with fractions and exponents. We hope that this article has helped you to better understand how to simplify expressions using the distributive property.

Practice Problems

1. Simplify the expression $(3x + 2y)(2x - 3y)$.
2. Simplify the expression $(4a - 2b)(3a + 2b)$.
3. Simplify the expression $(2c + 3d)(c - 2d)$.

Step-by-Step Solutions

1. Simplify the expression $(3x + 2y)(2x - 3y)$.

Using the distributive property, we can rewrite the expression as:

$(3x + 2y)(2x - 3y) = 3x(2x) + 3x(-3y) + 2y(2x) + 2y(-3y)$

Now, we can simplify each term:

$3x(2x) = 6x^2$

$3x(-3y) = -9xy$

$2y(2x) = 4xy$

$2y(-3y) = -6y^2$

Now, we can combine like terms:

$6x^2 - 9xy + 4xy - 6y^2$

The terms $-9xy$ and $4xy$ cancel each other out, so the expression simplifies to:

$6x^2 - 6y^2$

2. Simplify the expression $(4a - 2b)(3a + 2b)$.

Using the distributive property, we can rewrite the expression as:

$(4a - 2b)(3a + 2b) = 4a(3a) + 4a(2b) - 2b(3a) - 2b(2b)$

Now, we can simplify each term:

$4a(3a) = 12a^2$

$4a(2b) = 8ab$

$-2b(3a) = -6ab$

$-2b(2b) = -4b^2$

Now, we can combine like terms:

$12a^2 + 8ab - 6ab - 4b^2$

The terms $8ab$ and $-6ab$ cancel each other out, so the expression simplifies to:

$12a^2 - 4b^2$

3. Simplify the expression $(2c + 3d)(c - 2d)$.

Using the distributive property, we can rewrite the expression as:

$(2c + 3d)(c - 2d) = 2c(c) + 2c(-2d) + 3d(c) + 3d(-2d)$

Now, we can simplify each term:

$2c(c) = 2c^2$

$2c(-2d) = -4cd$

$3d(c) = 3cd$

$3d(-2d) = -6d^2$

Now, we can combine like terms:

$2c^2 - 4cd + 3cd - 6d^2$

The terms $-4cd$ and $3cd$ cancel each other out, so the expression simplifies to:

$2c^2 - 6d^2$