Simplify The Expression: ( C 10 C 8 D 2 ) − 5 \left(\frac{c^{10}}{c^8 D^2}\right)^{-5} ( C 8 D 2 C 10 ​ ) − 5

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve complex problems and understand the underlying concepts. One of the most common techniques used to simplify expressions is exponent rules. In this article, we will focus on simplifying the given expression using exponent rules.

Understanding Exponent Rules

Before we dive into simplifying the expression, let's review some basic exponent rules. Exponents are a shorthand way of representing repeated multiplication. For example, $a^3$ means $a \times a \times a$. When we have a fraction with exponents, we can use the following rules to simplify it:

• When we have a fraction with the same base, we can subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$
• When we have a fraction with different bases, we can use the rule: $\frac{a^m}{b^n} = \frac{a^m}{1} \times \frac{1}{b^n} = \frac{a^m}{b^n}$
• When we have a power raised to a power, we can multiply the exponents: $(a^m)^n = a^{m \times n}$

Simplifying the Expression

Now that we have reviewed the exponent rules, let's simplify the given expression: $\left(\frac{c^{10}}{c^8 d^2}\right)^{-5}$.

To simplify this expression, we can start by using the rule for a power raised to a power: $(a^m)^n = a^{m \times n}$. In this case, we have a fraction raised to a power, so we can use the rule: $\left(\frac{a^m}{b^n}\right)^p = \frac{a^{m \times p}}{b^{n \times p}}$.

Applying this rule to the given expression, we get:

$\left(\frac{c^{10}}{c^8 d^2}\right)^{-5} = \frac{c^{10 \times -5}}{c^{8 \times -5} d^{2 \times -5}}$

Simplifying the Exponents

Now that we have applied the rule, let's simplify the exponents. When we multiply a negative exponent by a positive exponent, the result is a positive exponent. For example, $-5 \times 10 = -50$ and $-5 \times 8 = -40$.

So, the expression becomes:

$\frac{c^{-50}}{c^{-40} d^{-10}}$

Canceling Out the Common Factors

Now that we have simplified the exponents, let's cancel out the common factors. When we have a fraction with the same base, we can subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$.

In this case, we have the same base $c$ in the numerator and denominator, so we can subtract the exponents:

$\frac{c^{-50}}{c^{-40} d^{-10}} = c^{-50 + 40} d^{-10}$

Simplifying the Expression Further

Now that we have canceled out the common factors, let's simplify the expression further. When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, $a^{-m} = \frac{1}{a^m}$.

In this case, we have a negative exponent $-10$ on the variable $d$, so we can rewrite it as a positive exponent:

$c^{-10} d^{-10} = \frac{1}{c^{10} d^{10}}$

Conclusion

In this article, we simplified the expression $\left(\frac{c^{10}}{c^8 d^2}\right)^{-5}$ using exponent rules. We applied the rule for a power raised to a power, simplified the exponents, canceled out the common factors, and finally simplified the expression further.

The final simplified expression is $\frac{1}{c^{10} d^{10}}$. This expression can be further simplified by canceling out the common factors or by rewriting it in a different form.

Final Answer

The final answer is $\boxed{\frac{1}{c^{10} d^{10}}}$.

Introduction

In our previous article, we simplified the expression $\left(\frac{c^{10}}{c^8 d^2}\right)^{-5}$ using exponent rules. In this article, we will answer some frequently asked questions related to the simplification of this expression.

Q&A

Q: What is the rule for a power raised to a power?

A: The rule for a power raised to a power is: $(a^m)^n = a^{m \times n}$. This rule can be used to simplify expressions with exponents.

Q: How do I simplify a fraction with exponents?

A: To simplify a fraction with exponents, you can use the following rules:

• When you have a fraction with the same base, you can subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$
• When you have a fraction with different bases, you can use the rule: $\frac{a^m}{b^n} = \frac{a^m}{1} \times \frac{1}{b^n} = \frac{a^m}{b^n}$
• When you have a power raised to a power, you can multiply the exponents: $(a^m)^n = a^{m \times n}$

Q: What is the difference between a positive and negative exponent?

A: A positive exponent represents a power that is raised to a positive value, while a negative exponent represents a power that is raised to a negative value. For example, $a^3$ is a positive exponent, while $a^{-3}$ is a negative exponent.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can rewrite it as a positive exponent by taking the reciprocal of the base. For example, $a^{-m} = \frac{1}{a^m}$.

Q: Can I cancel out common factors in an expression with exponents?

A: Yes, you can cancel out common factors in an expression with exponents. When you have a fraction with the same base, you can subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$.

Q: What is the final simplified expression for $\left(\frac{c^{10}}{c^8 d^2}\right)^{-5}$?

A: The final simplified expression for $\left(\frac{c^{10}}{c^8 d^2}\right)^{-5}$ is $\frac{1}{c^{10} d^{10}}$.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression $\left(\frac{c^{10}}{c^8 d^2}\right)^{-5}$. We covered topics such as exponent rules, simplifying fractions with exponents, and canceling out common factors.

Final Answer

The final answer is $\boxed{\frac{1}{c^{10} d^{10}}}$.