Simplify The Expression:${ (-2y)\left(-3x 2y 2\right)\left(-6xy^2\right) = }$

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Introduction

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In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. One common type of expression that requires simplification is the product of multiple terms, each containing variables and coefficients. In this article, we will focus on simplifying the expression $(-2y)\left(-3x^2y^2\right)\left(-6xy^2\right)$, which involves multiplying negative terms.

Understanding Negative Terms

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Before we dive into simplifying the expression, let's understand what negative terms are. A negative term is a term that has a negative coefficient. In the given expression, each term has a negative coefficient: $-2y$, $-3x^2y^2$, and $-6xy^2$. When we multiply negative terms, we need to remember that two negative numbers multiplied together result in a positive number.

Multiplying Negative Terms

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Now that we understand negative terms, let's multiply the given expression. To simplify the expression, we need to multiply the coefficients and variables separately.

$$(-2y)\left(-3x^2y^2\right)\left(-6xy^2\right) = (-2) \cdot (-3) \cdot (-6) \cdot (x^2) \cdot (y^2) \cdot (x) \cdot (y^2)$$

Simplifying the Coefficients

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When we multiply the coefficients, we need to remember that two negative numbers multiplied together result in a positive number. Therefore, the product of the coefficients is:

$$(-2) \cdot (-3) \cdot (-6) = -36$$

Simplifying the Variables

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When we multiply the variables, we need to add their exponents. In this case, we have:

$$x^2 \cdot x = x^{2+1} = x^3$$

$$y^2 \cdot y^2 = y^{2+2} = y^4$$

Combining the Simplified Coefficients and Variables

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Now that we have simplified the coefficients and variables, we can combine them to get the final simplified expression.

$$(-2y)\left(-3x^2y^2\right)\left(-6xy^2\right) = -36x^3y^4$$

Conclusion

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In this article, we simplified the expression $(-2y)\left(-3x^2y^2\right)\left(-6xy^2\right)$ by multiplying negative terms. We learned that two negative numbers multiplied together result in a positive number, and we applied this rule to simplify the coefficients. We also learned how to simplify variables by adding their exponents. By following these steps, we arrived at the final simplified expression: $-36x^3y^4$.

Frequently Asked Questions

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Q: What is the rule for multiplying negative terms?

A: The rule for multiplying negative terms is that two negative numbers multiplied together result in a positive number.

Q: How do we simplify variables when multiplying terms?

A: When multiplying variables, we add their exponents.

Q: What is the final simplified expression for the given problem?

A: The final simplified expression is $-36x^3y^4$.

Example Problems

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Problem 1: Simplify the expression (-4x)\left(-2y^2\right)\left(-3z^3)

To simplify this expression, we need to multiply the coefficients and variables separately.

$$(-4x)\left(-2y^2\right)\left(-3z^3\right) = (-4) \cdot (-2) \cdot (-3) \cdot (x) \cdot (y^2) \cdot (z^3)$$

When we multiply the coefficients, we get:

$$(-4) \cdot (-2) \cdot (-3) = -24$$

When we multiply the variables, we get:

$$x \cdot y^2 \cdot z^3 = x \cdot y^2 \cdot z^3$$

Therefore, the final simplified expression is:

$$(-4x)\left(-2y^2\right)\left(-3z^3\right) = -24x \cdot y^2 \cdot z^3$$

Problem 2: Simplify the expression (-3y^2)\left(-2x^3\right)\left(-4z^4)

To simplify this expression, we need to multiply the coefficients and variables separately.

$$(-3y^2)\left(-2x^3\right)\left(-4z^4\right) = (-3) \cdot (-2) \cdot (-4) \cdot (y^2) \cdot (x^3) \cdot (z^4)$$

When we multiply the coefficients, we get:

$$(-3) \cdot (-2) \cdot (-4) = -24$$

When we multiply the variables, we get:

$$y^2 \cdot x^3 \cdot z^4 = y^2 \cdot x^3 \cdot z^4$$

Therefore, the final simplified expression is:

$$(-3y^2)\left(-2x^3\right)\left(-4z^4\right) = -24y^2 \cdot x^3 \cdot z^4$$

Final Thoughts

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Simplifying expressions is an essential skill in algebra that helps us solve equations and inequalities. By understanding negative terms and how to multiply them, we can simplify complex expressions and arrive at the final answer. In this article, we simplified the expression $(-2y)\left(-3x^2y^2\right)\left(-6xy^2\right)$ by multiplying negative terms and arrived at the final simplified expression: $-36x^3y^4$. We also provided example problems to help you practice simplifying expressions.

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Introduction

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In our previous article, we simplified the expression $(-2y)\left(-3x^2y^2\right)\left(-6xy^2\right)$ by multiplying negative terms. We learned that two negative numbers multiplied together result in a positive number, and we applied this rule to simplify the coefficients. We also learned how to simplify variables by adding their exponents. In this article, we will answer some frequently asked questions about simplifying expressions.

Q&A

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Q: What is the rule for multiplying negative terms?

A: The rule for multiplying negative terms is that two negative numbers multiplied together result in a positive number.

Q: How do we simplify variables when multiplying terms?

A: When multiplying variables, we add their exponents.

Q: What is the final simplified expression for the given problem?

A: The final simplified expression is $-36x^3y^4$.

Q: Can we simplify expressions with variables that have the same base but different exponents?

A: Yes, we can simplify expressions with variables that have the same base but different exponents. We add their exponents when multiplying variables.

Q: How do we simplify expressions with variables that have different bases?

A: When simplifying expressions with variables that have different bases, we cannot add their exponents. Instead, we leave the variables as they are.

Q: Can we simplify expressions with variables that have negative exponents?

A: Yes, we can simplify expressions with variables that have negative exponents. We can rewrite the variable with a negative exponent as a fraction.

Q: How do we simplify expressions with variables that have fractional exponents?

A: When simplifying expressions with variables that have fractional exponents, we can rewrite the variable with a fractional exponent as a product of a variable with an integer exponent and a variable with a fractional exponent.

Q: Can we simplify expressions with variables that have imaginary exponents?

A: Yes, we can simplify expressions with variables that have imaginary exponents. We can rewrite the variable with an imaginary exponent as a product of a variable with a real exponent and a variable with an imaginary exponent.

Example Problems

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Problem 1: Simplify the expression $(-4x)\left(-2y^2\right)\left(-3z^3\right)$

To simplify this expression, we need to multiply the coefficients and variables separately.

$$(-4x)\left(-2y^2\right)\left(-3z^3\right) = (-4) \cdot (-2) \cdot (-3) \cdot (x) \cdot (y^2) \cdot (z^3)$$

When we multiply the coefficients, we get:

$$(-4) \cdot (-2) \cdot (-3) = -24$$

When we multiply the variables, we get:

$$x \cdot y^2 \cdot z^3 = x \cdot y^2 \cdot z^3$$

Therefore, the final simplified expression is:

$$(-4x)\left(-2y^2\right)\left(-3z^3\right) = -24x \cdot y^2 \cdot z^3$$

Problem 2: Simplify the expression $(-3y^2)\left(-2x^3\right)\left(-4z^4\right)$

To simplify this expression, we need to multiply the coefficients and variables separately.

$$(-3y^2)\left(-2x^3\right)\left(-4z^4\right) = (-3) \cdot (-2) \cdot (-4) \cdot (y^2) \cdot (x^3) \cdot (z^4)$$

When we multiply the coefficients, we get:

$$(-3) \cdot (-2) \cdot (-4) = -24$$

When we multiply the variables, we get:

$$y^2 \cdot x^3 \cdot z^4 = y^2 \cdot x^3 \cdot z^4$$

Therefore, the final simplified expression is:

$$(-3y^2)\left(-2x^3\right)\left(-4z^4\right) = -24y^2 \cdot x^3 \cdot z^4$$

Final Thoughts

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Simplifying expressions is an essential skill in algebra that helps us solve equations and inequalities. By understanding negative terms and how to multiply them, we can simplify complex expressions and arrive at the final answer. In this article, we answered some frequently asked questions about simplifying expressions and provided example problems to help you practice simplifying expressions.

Additional Resources

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• Algebra Tutorial
• Simplifying Expressions
• Algebra Practice Problems

Conclusion

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Simplifying expressions is a crucial skill in algebra that helps us solve equations and inequalities. By understanding negative terms and how to multiply them, we can simplify complex expressions and arrive at the final answer. In this article, we answered some frequently asked questions about simplifying expressions and provided example problems to help you practice simplifying expressions. We hope this article has been helpful in your understanding of simplifying expressions.