Use The Distributive Property To Remove The Parentheses:$\[ 8z^9(9z + 4z^6 + 2) \\]

Feb 28, 2025 by ADMIN 84 views

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Introduction

The distributive property is a fundamental concept in algebra that allows us to remove parentheses from expressions by multiplying each term inside the parentheses with the factor outside. In this article, we will explore how to use the distributive property to remove the parentheses from the given expression: $8z^9(9z + 4z^6 + 2)$ . We will break down the process step by step and provide examples to illustrate the concept.

Understanding the Distributive Property

The distributive property states that for any real numbers $a$ , $b$ , and $c$ , the following equation holds:

$a(b + c) = ab + ac$

This means that we can remove the parentheses by multiplying each term inside the parentheses with the factor outside. In other words, we can distribute the factor $a$ to each term inside the parentheses.

Applying the Distributive Property

Now, let's apply the distributive property to the given expression: $8z^9(9z + 4z^6 + 2)$ . We can start by multiplying the factor $8z^9$ with each term inside the parentheses:

$8z^9(9z + 4z^6 + 2) = 8z^9(9z) + 8z^9(4z^6) + 8z^9(2)$

Simplifying the Expression

Now, let's simplify each term by multiplying the factors:

$8z^9(9z) = 72z^{10}$

$8z^9(4z^6) = 32z^{15}$

$8z^9(2) = 16z^{18}$

Combining the Terms

Now, let's combine the terms to get the final expression:

$72z^{10} + 32z^{15} + 16z^{18}$

Conclusion

In this article, we used the distributive property to remove the parentheses from the given expression: $8z^9(9z + 4z^6 + 2)$ . We broke down the process step by step and provided examples to illustrate the concept. By applying the distributive property, we were able to simplify the expression and remove the parentheses.

Examples

Here are some examples of using the distributive property to remove parentheses:

$3(2x + 5) = 6x + 15$
$4(3y - 2) = 12y - 8$
$2(6z + 3) = 12z + 6$

Tips and Tricks

Here are some tips and tricks for using the distributive property:

Make sure to multiply each term inside the parentheses with the factor outside.
Simplify each term by multiplying the factors.
Combine the terms to get the final expression.

Practice Problems

Here are some practice problems for using the distributive property:

$5(2x + 3) = ?$
$3(4y - 2) = ?$
$2(6z + 4) = ?$

Solutions

Here are the solutions to the practice problems:

$5(2x + 3) = 10x + 15$
$3(4y - 2) = 12y - 6$
$2(6z + 4) = 12z + 8$

Final Thoughts

In conclusion, the distributive property is a powerful tool for simplifying expressions and removing parentheses. By applying the distributive property, we can break down complex expressions into simpler terms and make them easier to work with. With practice and patience, you can become proficient in using the distributive property and tackle even the most challenging algebra problems.

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Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to remove parentheses from expressions by multiplying each term inside the parentheses with the factor outside.

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply each term inside the parentheses with the factor outside. For example, if you have the expression $a(b + c)$ , you can apply the distributive property by multiplying $a$ with each term inside the parentheses: $ab + ac$ .

Q: What are some common mistakes to avoid when using the distributive property?

A: Some common mistakes to avoid when using the distributive property include:

Forgetting to multiply each term inside the parentheses with the factor outside.
Not simplifying each term by multiplying the factors.
Not combining the terms to get the final expression.

Q: Can I use the distributive property with negative numbers?

A: Yes, you can use the distributive property with negative numbers. For example, if you have the expression $-a(b + c)$ , you can apply the distributive property by multiplying $-a$ with each term inside the parentheses: $-ab - ac$ .

Q: Can I use the distributive property with fractions?

A: Yes, you can use the distributive property with fractions. For example, if you have the expression $\frac{a}{b}(c + d)$ , you can apply the distributive property by multiplying $\frac{a}{b}$ with each term inside the parentheses: $\frac{ac}{b} + \frac{ad}{b}$ .

Q: Can I use the distributive property with exponents?

A: Yes, you can use the distributive property with exponents. For example, if you have the expression $a^m(b + c)$ , you can apply the distributive property by multiplying $a^m$ with each term inside the parentheses: $a^mb + a^mc$ .

Q: How do I know when to use the distributive property?

A: You should use the distributive property whenever you have an expression with parentheses and you want to simplify it. The distributive property is a powerful tool for simplifying expressions and making them easier to work with.

Q: Can I use the distributive property with variables?

A: Yes, you can use the distributive property with variables. For example, if you have the expression $a(x + y)$ , you can apply the distributive property by multiplying $a$ with each term inside the parentheses: $ax + ay$ .

Q: Can I use the distributive property with coefficients?

A: Yes, you can use the distributive property with coefficients. For example, if you have the expression $2a(b + c)$ , you can apply the distributive property by multiplying $2a$ with each term inside the parentheses: $2ab + 2ac$ .

Q: How do I check my work when using the distributive property?

A: To check your work when using the distributive property, you can plug in some values for the variables and see if the expression simplifies correctly. You can also use a calculator or a computer algebra system to check your work.

Q: Can I use the distributive property with complex expressions?

A: Yes, you can use the distributive property with complex expressions. For example, if you have the expression $a(b + c + d)$ , you can apply the distributive property by multiplying $a$ with each term inside the parentheses: $ab + ac + ad$ .

Q: Can I use the distributive property with expressions with multiple parentheses?

A: Yes, you can use the distributive property with expressions with multiple parentheses. For example, if you have the expression $a(b + c)(d + e)$ , you can apply the distributive property by multiplying $a$ with each term inside the first parentheses, and then multiplying the result with each term inside the second parentheses: $ab(d + e) + ac(d + e)$ .

Q: Can I use the distributive property with expressions with variables and constants?

A: Yes, you can use the distributive property with expressions with variables and constants. For example, if you have the expression $a(b + c) + d$ , you can apply the distributive property by multiplying $a$ with each term inside the parentheses, and then adding the result to $d$ : $ab + ac + d$ .

Q: Can I use the distributive property with expressions with exponents and variables?

A: Yes, you can use the distributive property with expressions with exponents and variables. For example, if you have the expression $a^m(b + c)$ , you can apply the distributive property by multiplying $a^m$ with each term inside the parentheses: $a^mb + a^mc$ .

Q: Can I use the distributive property with expressions with fractions and variables?

A: Yes, you can use the distributive property with expressions with fractions and variables. For example, if you have the expression $\frac{a}{b}(c + d)$ , you can apply the distributive property by multiplying $\frac{a}{b}$ with each term inside the parentheses: $\frac{ac}{b} + \frac{ad}{b}$ .

Q: Can I use the distributive property with expressions with negative numbers and variables?

A: Yes, you can use the distributive property with expressions with negative numbers and variables. For example, if you have the expression $-a(b + c)$ , you can apply the distributive property by multiplying $-a$ with each term inside the parentheses: $-ab - ac$ .

Q: Can I use the distributive property with expressions with coefficients and variables?

A: Yes, you can use the distributive property with expressions with coefficients and variables. For example, if you have the expression $2a(b + c)$ , you can apply the distributive property by multiplying $2a$ with each term inside the parentheses: $2ab + 2ac$ .

Q: Can I use the distributive property with expressions with exponents and coefficients?

A: Yes, you can use the distributive property with expressions with exponents and coefficients. For example, if you have the expression $a^m(2b + c)$ , you can apply the distributive property by multiplying $a^m$ with each term inside the parentheses: $2a^mb + a^mc$ .

Q: Can I use the distributive property with expressions with fractions and coefficients?

A: Yes, you can use the distributive property with expressions with fractions and coefficients. For example, if you have the expression $\frac{a}{b}(2c + d)$ , you can apply the distributive property by multiplying $\frac{a}{b}$ with each term inside the parentheses: $\frac{2ac}{b} + \frac{ad}{b}$ .

Q: Can I use the distributive property with expressions with negative numbers and coefficients?

A: Yes, you can use the distributive property with expressions with negative numbers and coefficients. For example, if you have the expression $-2a(b + c)$ , you can apply the distributive property by multiplying $-2a$ with each term inside the parentheses: $-2ab - 2ac$ .

Q: Can I use the distributive property with expressions with exponents and fractions?

A: Yes, you can use the distributive property with expressions with exponents and fractions. For example, if you have the expression $a^m(\frac{b}{c} + d)$ , you can apply the distributive property by multiplying $a^m$ with each term inside the parentheses: $\frac{a^mb}{c} + a^md$ .

Q: Can I use the distributive property with expressions with variables and fractions?

A: Yes, you can use the distributive property with expressions with variables and fractions. For example, if you have the expression $a(\frac{b}{c} + d)$ , you can apply the distributive property by multiplying $a$ with each term inside the parentheses: $\frac{ab}{c} + ad$ .

Q: Can I use the distributive property with expressions with negative numbers and fractions?

A: Yes, you can use the distributive property with expressions with negative numbers and fractions. For example, if you have the expression $-a(\frac{b}{c} + d)$ , you can apply the distributive property by multiplying $-a$ with each term inside the parentheses: $-\frac{ab}{c} - ad$ .

Q: Can I use the distributive property with expressions with coefficients and fractions?

A: Yes, you can use the distributive property with expressions with coefficients and fractions. For example, if you have the expression $2a(\frac{b}{c} + d)$ , you can apply the distributive property by multiplying $2a$ with each term inside the parentheses: $\frac{2ab}{c} + 2ad$ .

Q: Can I use the distributive property with expressions with exponents and coefficients?

A: Yes, you can use the distributive property with expressions with exponents and coefficients. For example, if you have the expression $a^m(2b + c)$ , you can apply the distributive property by