Simplify The Expression: ${ \left(\frac{-2 A^3 B^2 C^0}{3 A^2 B^3 C 7}\right) {-2} }$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of the rules of exponents and fractions. In this article, we will focus on simplifying the given expression, which involves negative exponents and variables with different powers. We will break down the expression step by step, using the rules of exponents and fractions to simplify it.

Understanding the Expression

The given expression is $\left(\frac{-2 a^3 b^2 c^0}{3 a^2 b^3 c^7}\right)^{-2}$. This expression involves a fraction with variables and constants in the numerator and denominator. The variables are $a$, $b$, and $c$, and they have different powers. The expression is raised to the power of $-2$, which means we need to apply the rule of negative exponents.

Rule of Negative Exponents

The rule of negative exponents states that for any non-zero number $a$ and any integer $n$, $a^{-n} = \frac{1}{a^n}$. This rule can be applied to the given expression to simplify it.

Simplifying the Expression

To simplify the expression, we need to apply the rule of negative exponents. We can start by rewriting the expression as $\frac{1}{\left(\frac{-2 a^3 b^2 c^0}{3 a^2 b^3 c^7}\right)^2}$.

Applying the Rule of Negative Exponents

Now, we can apply the rule of negative exponents to the expression. We can rewrite the expression as $\frac{1}{\left(\frac{-2 a^3 b^2 c^0}{3 a^2 b^3 c^7}\right)^2} = \frac{1}{\frac{(-2 a^3 b^2 c^0)^2}{(3 a^2 b^3 c^7)^2}}$.

Simplifying the Numerator and Denominator

We can simplify the numerator and denominator separately. The numerator is $(-2 a^3 b^2 c^0)^2$, and the denominator is $(3 a^2 b^3 c^7)^2$.

Simplifying the Numerator

The numerator is $(-2 a^3 b^2 c^0)^2$. We can simplify this expression by applying the rule of exponents, which states that for any non-zero number $a$ and any integers $m$ and $n$, $(a^m)^n = a^{m \cdot n}$. Applying this rule, we get $(-2 a^3 b^2 c^0)^2 = (-2)^2 (a^3)^2 (b^2)^2 (c^0)^2$.

Simplifying the Denominator

The denominator is $(3 a^2 b^3 c^7)^2$. We can simplify this expression by applying the rule of exponents, which states that for any non-zero number $a$ and any integers $m$ and $n$, $(a^m)^n = a^{m \cdot n}$. Applying this rule, we get $(3 a^2 b^3 c^7)^2 = (3)^2 (a^2)^2 (b^3)^2 (c^7)^2$.

Combining the Numerator and Denominator

Now, we can combine the numerator and denominator to get the final simplified expression. We get $\frac{1}{\frac{(-2)^2 (a^3)^2 (b^2)^2 (c^0)^2}{(3)^2 (a^2)^2 (b^3)^2 (c^7)^2}}$.

Simplifying the Expression Further

We can simplify the expression further by canceling out common factors in the numerator and denominator. We get $\frac{1}{\frac{4 a^6 b^4 c^0}{9 a^4 b^6 c^{14}}}$.

Canceling Out Common Factors

We can cancel out common factors in the numerator and denominator. We get $\frac{1}{\frac{4 a^2 b^{-2} c^{-14}}{9}}$.

Simplifying the Expression

We can simplify the expression by multiplying the numerator and denominator by the reciprocal of the denominator. We get $\frac{9}{4 a^2 b^{-2} c^{-14}}$.

Simplifying the Expression Further

We can simplify the expression further by applying the rule of negative exponents. We get $\frac{9}{4 a^2 \frac{1}{b^2} \frac{1}{c^{14}}}$.

Simplifying the Expression

We can simplify the expression by rewriting it as $\frac{9 c^{14}}{4 a^2 b^2}$.

Conclusion

In this article, we simplified the given expression $\left(\frac{-2 a^3 b^2 c^0}{3 a^2 b^3 c^7}\right)^{-2}$ using the rules of exponents and fractions. We applied the rule of negative exponents to simplify the expression, and then we canceled out common factors in the numerator and denominator. Finally, we simplified the expression further by applying the rule of negative exponents and rewriting it in a simpler form.

Final Answer

The final simplified expression is $\frac{9 c^{14}}{4 a^2 b^2}$.

Key Takeaways

• The rule of negative exponents states that for any non-zero number $a$ and any integer $n$, $a^{-n} = \frac{1}{a^n}$.
• To simplify an expression with negative exponents, we can apply the rule of negative exponents and then cancel out common factors in the numerator and denominator.
• The rule of exponents states that for any non-zero number $a$ and any integers $m$ and $n$, $(a^m)^n = a^{m \cdot n}$.
• We can simplify an expression by rewriting it in a simpler form using the rules of exponents and fractions.
  

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Introduction

In our previous article, we simplified the given expression $\left(\frac{-2 a^3 b^2 c^0}{3 a^2 b^3 c^7}\right)^{-2}$ using the rules of exponents and fractions. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

Q: What is the rule of negative exponents?

A: The rule of negative exponents states that for any non-zero number $a$ and any integer $n$, $a^{-n} = \frac{1}{a^n}$.

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

A: To simplify an expression with negative exponents, you can apply the rule of negative exponents and then cancel out common factors in the numerator and denominator.

Q: What is the rule of exponents?

A: The rule of exponents states that for any non-zero number $a$ and any integers $m$ and $n$, $(a^m)^n = a^{m \cdot n}$.

Q: How do I simplify an expression using the rule of exponents?

A: To simplify an expression using the rule of exponents, you can apply the rule by multiplying the exponents of the same base.

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{9 c^{14}}{4 a^2 b^2}$.

Q: Can I simplify the expression further?

A: Yes, you can simplify the expression further by canceling out common factors in the numerator and denominator.

Q: How do I cancel out common factors in the numerator and denominator?

A: To cancel out common factors in the numerator and denominator, you can divide both the numerator and denominator by the common factor.

Q: What is the importance of simplifying expressions?

A: Simplifying expressions is important because it helps to make the expression easier to understand and work with. It also helps to avoid errors and make calculations more efficient.

Q: Can I use the rule of negative exponents to simplify expressions with variables?

A: Yes, you can use the rule of negative exponents to simplify expressions with variables. The rule applies to any non-zero number, including variables.

Q: How do I apply the rule of negative exponents to expressions with variables?

A: To apply the rule of negative exponents to expressions with variables, you can follow the same steps as you would with numerical expressions.

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

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

• Not applying the rule of negative exponents correctly
• Not canceling out common factors in the numerator and denominator
• Not following the order of operations
• Not simplifying the expression further when possible

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression $\left(\frac{-2 a^3 b^2 c^0}{3 a^2 b^3 c^7}\right)^{-2}$. We covered topics such as the rule of negative exponents, the rule of exponents, and how to simplify expressions using these rules. We also discussed the importance of simplifying expressions and some common mistakes to avoid.

Final Answer

The final simplified expression is $\frac{9 c^{14}}{4 a^2 b^2}$.

Key Takeaways

• The rule of negative exponents states that for any non-zero number $a$ and any integer $n$, $a^{-n} = \frac{1}{a^n}$.
• To simplify an expression with negative exponents, you can apply the rule of negative exponents and then cancel out common factors in the numerator and denominator.
• The rule of exponents states that for any non-zero number $a$ and any integers $m$ and $n$, $(a^m)^n = a^{m \cdot n}$.
• Simplifying expressions is important because it helps to make the expression easier to understand and work with.
• Some common mistakes to avoid when simplifying expressions include not applying the rule of negative exponents correctly, not canceling out common factors in the numerator and denominator, and not following the order of operations.