Simplify The Expression:$ \left(x^7 Y^{-c} C\right)\left(-x^7 Y^7 C^x\right) = $

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve complex problems and understand the underlying concepts. In this article, we will focus on simplifying the given expression: $\left(x^7 y^{-c} c\right)\left(-x^7 y^7 c^x\right) = $. We will break down the expression step by step, using various mathematical techniques and rules to simplify it.

Understanding the Expression

The given expression is a product of two terms: $\left(x^7 y^{-c} c\right)$ and $\left(-x^7 y^7 c^x\right)$. To simplify this expression, we need to apply the rules of exponents and algebraic manipulation.

Step 1: Apply the Product Rule

The product rule states that when we multiply two terms with the same base, we add their exponents. In this case, we have two terms with the base $x$: $x^7$ and $-x^7$. We can apply the product rule to simplify this part of the expression:

$$\left(x^7 y^{-c} c\right)\left(-x^7 y^7 c^x\right) = \left(x^{7+(-7)} y^{-c+7} c^{1+x}\right)$$

Step 2: Simplify the Exponents

Now that we have applied the product rule, we can simplify the exponents. We have $x^{7+(-7)}$, which simplifies to $x^0$. We also have $y^{-c+7}$, which can be simplified using the rule for negative exponents:

$$y^{-c+7} = \frac{y^7}{y^c}$$

Step 3: Simplify the Fraction

We now have a fraction: $\frac{y^7}{y^c}$. We can simplify this fraction by applying the rule for dividing like bases:

$$\frac{y^7}{y^c} = y^{7-c}$$

Step 4: Simplify the Expression

Now that we have simplified the exponents and the fraction, we can simplify the entire expression:

$$\left(x^7 y^{-c} c\right)\left(-x^7 y^7 c^x\right) = \left(x^0 y^{7-c} c^{1+x}\right)$$

Step 5: Simplify the Final Expression

We now have a simplified expression: $\left(x^0 y^{7-c} c^{1+x}\right)$. We can simplify this expression further by applying the rule for $x^0$:

$$x^0 = 1$$

So, the final simplified expression is:

$$\left(y^{7-c} c^{1+x}\right)$$

Conclusion

In this article, we have simplified the given expression: $\left(x^7 y^{-c} c\right)\left(-x^7 y^7 c^x\right) = $. We have applied various mathematical techniques and rules to simplify the expression, including the product rule, the rule for negative exponents, and the rule for dividing like bases. The final simplified expression is $\left(y^{7-c} c^{1+x}\right)$.

Tips and Tricks

• When simplifying expressions, always start by applying the product rule.
• Use the rule for negative exponents to simplify expressions with negative exponents.
• Apply the rule for dividing like bases to simplify fractions with like bases.
• Use the rule for $x^0$ to simplify expressions with $x^0$.

Common Mistakes

• Failing to apply the product rule when simplifying expressions.
• Not using the rule for negative exponents when simplifying expressions with negative exponents.
• Not applying the rule for dividing like bases when simplifying fractions with like bases.
• Not using the rule for $x^0$ when simplifying expressions with $x^0$.

Real-World Applications

Simplifying expressions is a crucial skill in mathematics, and it has many real-world applications. For example, in physics, simplifying expressions is used to solve complex problems in mechanics, electromagnetism, and thermodynamics. In engineering, simplifying expressions is used to design and analyze complex systems, such as electrical circuits and mechanical systems. In economics, simplifying expressions is used to model and analyze complex economic systems.

Final Thoughts

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Q&A: Simplifying Expressions

Q: What is the product rule in mathematics?

A: The product rule is a mathematical rule that states when we multiply two terms with the same base, we add their exponents. For example, if we have $x^a \cdot x^b$, the product rule tells us that we can simplify it to $x^{a+b}$.

Q: How do I simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, we can use the rule for negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. For example, if we have $x^{-2}$, we can simplify it to $\frac{1}{x^2}$.

Q: What is the rule for dividing like bases?

A: The rule for dividing like bases states that when we divide two terms with the same base, we subtract their exponents. For example, if we have $\frac{x^a}{x^b}$, the rule tells us that we can simplify it to $x^{a-b}$.

Q: How do I simplify expressions with $x^0$?

A: To simplify expressions with $x^0$, we can use the rule that $x^0 = 1$. For example, if we have $x^0 \cdot y^2$, we can simplify it to $y^2$.

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

A: Some common mistakes to avoid when simplifying expressions include:

• Failing to apply the product rule when simplifying expressions.
• Not using the rule for negative exponents when simplifying expressions with negative exponents.
• Not applying the rule for dividing like bases when simplifying fractions with like bases.
• Not using the rule for $x^0$ when simplifying expressions with $x^0$.

Q: How do I apply the product rule to simplify expressions?

A: To apply the product rule to simplify expressions, follow these steps:

1. Identify the terms with the same base.
2. Add their exponents.
3. Simplify the resulting expression.

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

A: Simplifying expressions has many real-world applications, including:

• Physics: Simplifying expressions is used to solve complex problems in mechanics, electromagnetism, and thermodynamics.
• Engineering: Simplifying expressions is used to design and analyze complex systems, such as electrical circuits and mechanical systems.
• Economics: Simplifying expressions is used to model and analyze complex economic systems.

Q: How do I know when to use the rule for negative exponents?

A: You should use the rule for negative exponents when you have an expression with a negative exponent, such as $x^{-2}$. The rule states that $a^{-n} = \frac{1}{a^n}$, so you can simplify the expression by rewriting it as a fraction.

Q: What is the difference between the product rule and the rule for dividing like bases?

A: The product rule and the rule for dividing like bases are two different rules that are used to simplify expressions. The product rule states that when you multiply two terms with the same base, you add their exponents. The rule for dividing like bases states that when you divide two terms with the same base, you subtract their exponents.

Q: How do I simplify expressions with multiple variables?

A: To simplify expressions with multiple variables, follow these steps:

1. Identify the terms with the same base.
2. Add or subtract their exponents, depending on whether you are multiplying or dividing.
3. Simplify the resulting expression.

Q: What are some tips for simplifying expressions?

A: Some tips for simplifying expressions include:

• Start by applying the product rule.
• Use the rule for negative exponents to simplify expressions with negative exponents.
• Apply the rule for dividing like bases to simplify fractions with like bases.
• Use the rule for $x^0$ to simplify expressions with $x^0$.
• Simplify expressions step by step, rather than trying to simplify them all at once.

Conclusion

Simplifying expressions is a fundamental skill in mathematics, and it has many real-world applications. By applying various mathematical techniques and rules, we can simplify complex expressions and solve complex problems. In this article, we have provided a comprehensive guide to simplifying expressions, including the product rule, the rule for negative exponents, and the rule for dividing like bases. We have also provided tips and tricks for simplifying expressions, as well as common mistakes to avoid.