Multiply And Simplify The Expression:$ 6 X Y^3 Z \cdot 3 X^2 Y X^2 }$Choose The Correct Simplified Form A. ${$18(x Y Z)^{10 $}$B. ${ 9 X^5 Y^4 Z\$} C. ${ 18 X^4 Y^3 Z\$} D. ${ 18 X^5 Y^4 Z\$}

Mar 9, 2025 by ADMIN 193 views

**Multiplying and Simplifying Algebraic Expressions**

Understanding the Basics of Algebraic Multiplication

Algebraic expressions are a fundamental concept in mathematics, and understanding how to multiply and simplify them is crucial for solving various mathematical problems. In this article, we will focus on multiplying and simplifying the given expression: $6 x y^3 z \cdot 3 x^2 y x^2$ . We will break down the process step by step and explore the different options for simplifying the expression.

Step 1: Multiply the Coefficients

The first step in multiplying the given expression is to multiply the coefficients. The coefficients are the numerical values associated with each variable in the expression. In this case, the coefficients are 6 and 3.

${ 6 \cdot 3 = 18 }$

So, the product of the coefficients is 18.

Step 2: Multiply the Variables

The next step is to multiply the variables. When multiplying variables, we add the exponents of the same variables. In this case, we have $x$ , $y$ , and $z$ .

${ x \cdot x^2 \cdot x^2 = x^{1+2+2} = x^5 }$

${ y^3 \cdot y \cdot y = y^{3+1+1} = y^5 }$

However, we notice that the exponent of $y$ in the original expression is $y^3$ , not $y^5$ . Therefore, the correct exponent for $y$ is $y^4$ .

${ y^3 \cdot y \cdot y = y^{3+1+1} = y^5 }$

However, we notice that the exponent of $y$ in the original expression is $y^3$ , not $y^5$ . Therefore, the correct exponent for $y$ is $y^4$ .

Step 3: Simplify the Expression

Now that we have multiplied the coefficients and variables, we can simplify the expression by combining the results.

${ 18 \cdot x^5 \cdot y^4 \cdot z }$

This is the simplified form of the given expression.

Comparing the Options

Now that we have simplified the expression, let's compare it with the given options.

A. $18(x y z)^{10}$

B. $9 x^5 y^4 z$

C. $18 x^4 y^3 z$

D. $18 x^5 y^4 z$

We can see that option D matches our simplified expression.

Conclusion

In conclusion, the correct simplified form of the given expression is $18 x^5 y^4 z$ . This is achieved by multiplying the coefficients and variables, and then simplifying the expression by combining the results.

Key Takeaways

When multiplying algebraic expressions, we multiply the coefficients and variables separately.
When multiplying variables, we add the exponents of the same variables.
The simplified form of the expression is $18 x^5 y^4 z$ .

Practice Problems

Multiply and simplify the expression: $2 x^2 y^3 z \cdot 4 x y^2 z$
Multiply and simplify the expression: $3 x^2 y^4 z \cdot 2 x y^3 z$
Multiply and simplify the expression: $5 x^3 y^2 z \cdot 3 x^2 y^3 z$

Answer Key

$8 x^3 y^5 z^2$
$6 x^3 y^7 z^2$
$15 x^5 y^5 z^2$

References

About the Author

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions related to multiplying and simplifying algebraic expressions.

Q: What is the order of operations when multiplying algebraic expressions?

A: When multiplying algebraic expressions, we follow the order of operations:

Multiply the coefficients
Multiply the variables
Simplify the expression by combining the results

Q: How do I multiply variables with exponents?

A: When multiplying variables with exponents, we add the exponents of the same variables. For example:

${ x^2 \cdot x^3 = x^{2+3} = x^5 }$

Q: What is the difference between multiplying and adding algebraic expressions?

A: When multiplying algebraic expressions, we multiply the coefficients and variables separately. When adding algebraic expressions, we combine like terms by adding or subtracting the coefficients of the same variables.

Q: Can I simplify an expression by canceling out variables?

A: Yes, you can simplify an expression by canceling out variables. For example:

${ \frac{x^2 y^3}{x^2 y^2} = \frac{x^2}{x^2} \cdot \frac{y^3}{y^2} = 1 \cdot y = y }$

Q: How do I simplify an expression with negative exponents?

A: When simplifying an expression with negative exponents, we can rewrite the expression with positive exponents by moving the variable to the other side of the fraction. For example:

${ \frac{1}{x^2} = x^{-2} }$

Q: Can I simplify an expression with fractions?

A: Yes, you can simplify an expression with fractions by multiplying the numerator and denominator by the same value. For example:

${ \frac{2x}{3y} \cdot \frac{3y}{2x} = \frac{2x \cdot 3y}{3y \cdot 2x} = 1 }$

Q: How do I simplify an expression with parentheses?

A: When simplifying an expression with parentheses, we can use the distributive property to multiply the terms inside the parentheses. For example:

${ (x + y) \cdot (x + y) = x \cdot x + x \cdot y + y \cdot x + y \cdot y }$

Q: Can I simplify an expression with exponents and fractions?

A: Yes, you can simplify an expression with exponents and fractions by following the order of operations and using the rules for multiplying and adding fractions. For example:

${ \frac{x^2}{y^2} \cdot \frac{y^3}{x^3} = \frac{x^2 \cdot y^3}{y^2 \cdot x^3} = \frac{x^2}{x^3} \cdot \frac{y^3}{y^2} = \frac{1}{x} \cdot y = \frac{y}{x} }$

Conclusion

In conclusion, multiplying and simplifying algebraic expressions is a crucial skill in mathematics. By following the order of operations and using the rules for multiplying and adding variables, fractions, and exponents, you can simplify complex expressions and solve mathematical problems with ease.

Practice Problems

Multiply and simplify the expression: $2x^2y^3z \cdot 4x^2y^2z$
Multiply and simplify the expression: $3x^2y^4z \cdot 2x^2y^3z$
Multiply and simplify the expression: $5x^3y^2z \cdot 3x^2y^3z$

Answer Key

$8x^4y^5z^2$
$6x^4y^7z^2$
$15x^5y^5z^2$

References

About the Author

The author is a mathematics enthusiast with a passion for teaching and learning. They have a strong background in algebra and have experience in tutoring and mentoring students.