Simplify The Expression:$ -5x^5y^5(-6xy^3) $

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Understanding the Problem

When simplifying algebraic expressions, it's essential to apply the rules of exponents and multiplication. In this problem, we're given the expression $-5x^5y^5(-6xy^3)$, and we need to simplify it by applying the rules of exponents and multiplication.

Applying the Rules of Exponents

The rules of exponents state that when multiplying two numbers with the same base, we add their exponents. In this case, we have two numbers with the same base, $x$, and two numbers with the same base, $y$. We can apply the rule of exponents to simplify the expression.

Multiplying Negative Numbers

When multiplying two negative numbers, we get a positive result. In this case, we have two negative numbers, $-5$ and $-6$, which we will multiply together.

Distributing the Negative Sign

When a negative sign is distributed to a product, it changes the sign of each factor. In this case, we have a negative sign distributed to the product $-6xy^3$, which changes the sign of each factor.

Simplifying the Expression

Now that we've applied the rules of exponents and multiplication, we can simplify the expression.

Step 1: Multiply the Coefficients

The coefficients of the expression are $-5$ and $-6$. We multiply these two numbers together to get:

$$-5 \times -6 = 30$$

Step 2: Apply the Rule of Exponents

We have two numbers with the same base, $x$, and two numbers with the same base, $y$. We can apply the rule of exponents to simplify the expression.

$$x^5 \times x = x^{5+1} = x^6$$

$$y^5 \times y^3 = y^{5+3} = y^8$$

Step 3: Multiply the Terms

Now that we've applied the rule of exponents, we can multiply the terms together.

$$30x^6y^8 \times -6xy^3 = -180x^7y^{11}$$

Final Answer

The final answer is $-180x^7y^{11}$.

Conclusion

Simplifying algebraic expressions requires applying the rules of exponents and multiplication. In this problem, we applied the rules of exponents and multiplication to simplify the expression $-5x^5y^5(-6xy^3)$. We multiplied the coefficients together, applied the rule of exponents, and multiplied the terms together to get the final answer of $-180x^7y^{11}$.

Common Mistakes to Avoid

When simplifying algebraic expressions, it's essential to avoid common mistakes. Some common mistakes to avoid include:

• Not applying the rules of exponents correctly
• Not multiplying the coefficients together
• Not multiplying the terms together
• Not simplifying the expression correctly

Tips for Simplifying Algebraic Expressions

When simplifying algebraic expressions, it's essential to follow these tips:

• Apply the rules of exponents correctly
• Multiply the coefficients together
• Multiply the terms together
• Simplify the expression correctly
• Check your work for errors

Real-World Applications

Simplifying algebraic expressions has many real-world applications. Some examples include:

• Simplifying complex equations in physics and engineering
• Simplifying expressions in computer science and programming
• Simplifying expressions in finance and economics

Final Thoughts

Simplifying algebraic expressions is an essential skill in mathematics and has many real-world applications. By applying the rules of exponents and multiplication, we can simplify complex expressions and get the final answer. Remember to avoid common mistakes and follow the tips for simplifying algebraic expressions.

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Understanding the Problem

When simplifying algebraic expressions, it's essential to apply the rules of exponents and multiplication. In this problem, we're given the expression $-5x^5y^5(-6xy^3)$, and we need to simplify it by applying the rules of exponents and multiplication.

Q&A

Q: What are the rules of exponents?

A: The rules of exponents state that when multiplying two numbers with the same base, we add their exponents. For example, $x^2 \times x^3 = x^{2+3} = x^5$.

Q: How do we multiply negative numbers?

A: When multiplying two negative numbers, we get a positive result. For example, $-5 \times -6 = 30$.

Q: What is the rule for distributing a negative sign?

A: When a negative sign is distributed to a product, it changes the sign of each factor. For example, $-5 \times (2x + 3) = -10x - 15$.

Q: How do we simplify the expression $-5x^5y^5(-6xy^3)$?

A: To simplify the expression, we multiply the coefficients together, apply the rule of exponents, and multiply the terms together. The final answer is $-180x^7y^{11}$.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid include:

• Not applying the rules of exponents correctly
• Not multiplying the coefficients together
• Not multiplying the terms together
• Not simplifying the expression correctly

Q: What are some tips for simplifying algebraic expressions?

A: Some tips for simplifying algebraic expressions include:

• Apply the rules of exponents correctly
• Multiply the coefficients together
• Multiply the terms together
• Simplify the expression correctly
• Check your work for errors

Q: What are some real-world applications of simplifying algebraic expressions?

A: Some real-world applications of simplifying algebraic expressions include:

• Simplifying complex equations in physics and engineering
• Simplifying expressions in computer science and programming
• Simplifying expressions in finance and economics

Example Problems

Problem 1: Simplify the expression $2x^3y^2(3xy^4)$

A: To simplify the expression, we multiply the coefficients together, apply the rule of exponents, and multiply the terms together. The final answer is $6x^4y^6$.

Problem 2: Simplify the expression $-4x^2y^3(-2xy^2)$

A: To simplify the expression, we multiply the coefficients together, apply the rule of exponents, and multiply the terms together. The final answer is $8x^3y^5$.

Practice Problems

Problem 1: Simplify the expression $3x^2y^3(2xy^2)$

A: To simplify the expression, we multiply the coefficients together, apply the rule of exponents, and multiply the terms together.

Problem 2: Simplify the expression $-2x^3y^2(3xy^4)$

A: To simplify the expression, we multiply the coefficients together, apply the rule of exponents, and multiply the terms together.

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics and has many real-world applications. By applying the rules of exponents and multiplication, we can simplify complex expressions and get the final answer. Remember to avoid common mistakes and follow the tips for simplifying algebraic expressions.

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics and has many real-world applications. By practicing and mastering the rules of exponents and multiplication, we can simplify complex expressions and get the final answer.