Simplify The Expression: ( − 3 A B 4 ) ( − 2 B C ) ( − 5 C 3 (-3 A B^4)(-2 B C)(-5 C^3 ( − 3 A B 4 ) ( − 2 B C ) ( − 5 C 3 ]

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve complex problems and understand the underlying concepts. When dealing with algebraic expressions, we often encounter products of variables and constants. In this article, we will focus on simplifying the given expression: $(-3 a b^4)(-2 b c)(-5 c^3)$. We will use the properties of exponents, multiplication, and simplification to arrive at the final result.

Understanding the Expression

The given expression is a product of three terms: $(-3 a b^4)$, $(-2 b c)$, and $(-5 c^3)$. Each term contains variables and constants, and we need to simplify the expression by multiplying these terms together.

Multiplying the Terms

To simplify the expression, we will multiply the three terms together. When multiplying variables and constants, we follow the rule that states: $a^m \cdot a^n = a^{m+n}$. This means that when we multiply variables with the same base, we add their exponents.

Step 1: Multiply the First Two Terms

Let's start by multiplying the first two terms: $(-3 a b^4)$ and $(-2 b c)$. When multiplying variables, we multiply their coefficients and add their exponents.

import sympy as sp
a, b, c = sp.symbols('a b c')

term1 = -3 * a * b**4
term2 = -2 * b * c

result1 = sp.expand(term1 * term2)
print(result1)

Step 1 Result

The result of multiplying the first two terms is: $6 a b^5 c$.

Step 2: Multiply the Result with the Third Term

Now, let's multiply the result from Step 1 with the third term: $(-5 c^3)$. Again, we multiply the coefficients and add the exponents.

# Multiply the result with the third term
result2 = sp.expand(result1 * (-5 * c**3))
print(result2)

Step 2 Result

The result of multiplying the result from Step 1 with the third term is: $-30 a b^5 c^4$.

Conclusion

In this article, we simplified the given expression: $(-3 a b^4)(-2 b c)(-5 c^3)$. We used the properties of exponents, multiplication, and simplification to arrive at the final result. The simplified expression is: $-30 a b^5 c^4$.

Final Answer

The final answer is: $\boxed{-30 a b^5 c^4}$.

Frequently Asked Questions

• Q: What is the property of exponents used in this article? A: The property of exponents used in this article is: $a^m \cdot a^n = a^{m+n}$.
• Q: How do we multiply variables and constants? A: When multiplying variables and constants, we multiply their coefficients and add their exponents.
• Q: What is the simplified expression? A: The simplified expression is: $-30 a b^5 c^4$.

Further Reading

If you want to learn more about simplifying expressions, I recommend checking out the following resources:

• Khan Academy: Simplifying Expressions
• Mathway: Simplifying Expressions
• Wolfram Alpha: Simplifying Expressions

References

• [1] Khan Academy. (n.d.). Simplifying Expressions. Retrieved from https://www.khanacademy.org/math/algebra/x2f5f7c6/x2f5f7c7/x2f5f7c8
• [2] Mathway. (n.d.). Simplifying Expressions. Retrieved from https://www.mathway.com/subjects/algebra/simplifying-expressions
• [3] Wolfram Alpha. (n.d.). Simplifying Expressions. Retrieved from https://www.wolframalpha.com/input/?i=simplifying+expressions
  

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Introduction

In our previous article, we simplified the given expression: $(-3 a b^4)(-2 b c)(-5 c^3)$. We used the properties of exponents, multiplication, and simplification to arrive at the final result. In this article, we will answer some frequently asked questions related to simplifying expressions.

Q&A

Q: What is the property of exponents used in this article?

A: The property of exponents used in this article is: $a^m \cdot a^n = a^{m+n}$. This means that when we multiply variables with the same base, we add their exponents.

Q: How do we multiply variables and constants?

A: When multiplying variables and constants, we multiply their coefficients and add their exponents. For example, if we have the expression: $2x^2 \cdot 3x^3$, we would multiply the coefficients: $2 \cdot 3 = 6$, and add the exponents: $2 + 3 = 5$. The resulting expression would be: $6x^5$.

Q: What is the simplified expression?

A: The simplified expression is: $-30 a b^5 c^4$.

Q: How do we handle negative signs when multiplying expressions?

A: When multiplying expressions with negative signs, we multiply the coefficients and add the exponents, just like we do with positive signs. However, if the number of negative signs is odd, the result will be negative. If the number of negative signs is even, the result will be positive.

Q: Can we simplify expressions with variables in the denominator?

A: Yes, we can simplify expressions with variables in the denominator. To do this, we need to follow the rules of exponents and simplify the expression as much as possible.

Q: How do we handle fractions when simplifying expressions?

A: When simplifying expressions with fractions, we need to follow the rules of fractions and simplify the expression as much as possible. This may involve multiplying the numerator and denominator by the same value to eliminate the fraction.

Q: Can we simplify expressions with multiple variables?

A: Yes, we can simplify expressions with multiple variables. To do this, we need to follow the rules of exponents and simplify the expression as much as possible.

Q: How do we handle expressions with exponents and fractions?

A: When simplifying expressions with exponents and fractions, we need to follow the rules of exponents and fractions and simplify the expression as much as possible. This may involve multiplying the numerator and denominator by the same value to eliminate the fraction.

Examples

Example 1: Simplify the expression: $(-2x^2)(3x^3)$

A: To simplify this expression, we multiply the coefficients: $-2 \cdot 3 = -6$, and add the exponents: $2 + 3 = 5$. The resulting expression would be: $-6x^5$.

Example 2: Simplify the expression: $(2x^2)(-3x^3)$

A: To simplify this expression, we multiply the coefficients: $2 \cdot -3 = -6$, and add the exponents: $2 + 3 = 5$. The resulting expression would be: $-6x^5$.

Example 3: Simplify the expression: $(x^2)(-2x^3)$

A: To simplify this expression, we multiply the coefficients: $1 \cdot -2 = -2$, and add the exponents: $2 + 3 = 5$. The resulting expression would be: $-2x^5$.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions. We covered topics such as the property of exponents, multiplying variables and constants, handling negative signs, and simplifying expressions with variables in the denominator. We also provided examples to illustrate the concepts.

Final Answer

The final answer is: $\boxed{-30 a b^5 c^4}$.

Further Reading

If you want to learn more about simplifying expressions, I recommend checking out the following resources:

• Khan Academy: Simplifying Expressions
• Mathway: Simplifying Expressions
• Wolfram Alpha: Simplifying Expressions

References

• [1] Khan Academy. (n.d.). Simplifying Expressions. Retrieved from https://www.khanacademy.org/math/algebra/x2f5f7c6/x2f5f7c7/x2f5f7c8
• [2] Mathway. (n.d.). Simplifying Expressions. Retrieved from https://www.mathway.com/subjects/algebra/simplifying-expressions
• [3] Wolfram Alpha. (n.d.). Simplifying Expressions. Retrieved from https://www.wolframalpha.com/input/?i=simplifying+expressions