Evaluate The Expression: ( V − 3 ) − 1 \left(v^{-3}\right)^{-1} ( V − 3 ) − 1 .

Introduction

In mathematics, expressions involving exponents and powers are common and play a crucial role in various mathematical operations. One such expression is $\left(v^{-3}\right)^{-1}$, which involves the application of exponent rules to simplify the given expression. In this article, we will delve into the process of evaluating this expression and explore the underlying mathematical concepts.

Understanding Exponents and Powers

Before we proceed to evaluate the given expression, it is essential to understand the concept of exponents and powers. An exponent is a small number that is written to the upper right of a number or a variable, indicating how many times the base is multiplied by itself. For example, in the expression $v^3$, the exponent 3 indicates that the base $v$ is multiplied by itself three times, resulting in $v \times v \times v$.

Applying Exponent Rules

To evaluate the expression $\left(v^{-3}\right)^{-1}$, we need to apply the exponent rules. The first rule to apply is the power of a power rule, which states that for any numbers $a$ and $b$ and any integer $n$, $(a^b)^n = a^{b \cdot n}$. In this case, we have $(v^{-3})^{-1}$, which can be rewritten as $v^{(-3) \cdot (-1)}$.

Simplifying the Expression

Now that we have applied the power of a power rule, we can simplify the expression further. When we multiply two negative numbers, the result is a positive number. Therefore, $(-3) \cdot (-1) = 3$. So, the expression becomes $v^3$.

Evaluating the Final Expression

The final expression is $v^3$, which can be evaluated by multiplying the base $v$ by itself three times. Therefore, the value of the expression $\left(v^{-3}\right)^{-1}$ is $v \times v \times v = v^3$.

Conclusion

In conclusion, evaluating the expression $\left(v^{-3}\right)^{-1}$ involves applying the exponent rules, specifically the power of a power rule. By following these rules, we can simplify the expression and arrive at the final value of $v^3$. This process demonstrates the importance of understanding and applying exponent rules in mathematical operations.

Frequently Asked Questions

• What is the value of the expression $\left(v^{-3}\right)^{-1}$?
• How do we apply the exponent rules to simplify the expression?
• What is the final value of the expression $\left(v^{-3}\right)^{-1}$?

Answer

• The value of the expression $\left(v^{-3}\right)^{-1}$ is $v^3$.
• To apply the exponent rules, we use the power of a power rule, which states that for any numbers $a$ and $b$ and any integer $n$, $(a^b)^n = a^{b \cdot n}$.
• The final value of the expression $\left(v^{-3}\right)^{-1}$ is $v^3$.

Example Problems

• Evaluate the expression $\left(x^2\right)^{-3}$.
• Simplify the expression $\left(y^{-4}\right)^{-2}$.
• Evaluate the expression $\left(z^5\right)^{-1}$.

Solution

• To evaluate the expression $\left(x^2\right)^{-3}$, we apply the power of a power rule, resulting in $x^{2 \cdot (-3)} = x^{-6}$.
• To simplify the expression $\left(y^{-4}\right)^{-2}$, we apply the power of a power rule, resulting in $y^{(-4) \cdot (-2)} = y^8$.
• To evaluate the expression $\left(z^5\right)^{-1}$, we apply the power of a power rule, resulting in $z^{5 \cdot (-1)} = z^{-5}$.

Final Thoughts

In conclusion, evaluating the expression $\left(v^{-3}\right)^{-1}$ involves applying the exponent rules, specifically the power of a power rule. By following these rules, we can simplify the expression and arrive at the final value of $v^3$. This process demonstrates the importance of understanding and applying exponent rules in mathematical operations.

Introduction

In our previous article, we explored the process of evaluating the expression $\left(v^{-3}\right)^{-1}$, which involves applying the exponent rules to simplify the given expression. In this article, we will address some of the most frequently asked questions related to evaluating expressions with exponents and powers.

Q&A

Q: What is the value of the expression $\left(x^2\right)^{-3}$?

A: To evaluate the expression $\left(x^2\right)^{-3}$, we apply the power of a power rule, resulting in $x^{2 \cdot (-3)} = x^{-6}$.

Q: How do we simplify the expression $\left(y^{-4}\right)^{-2}$?

A: To simplify the expression $\left(y^{-4}\right)^{-2}$, we apply the power of a power rule, resulting in $y^{(-4) \cdot (-2)} = y^8$.

Q: What is the final value of the expression $\left(z^5\right)^{-1}$?

A: To evaluate the expression $\left(z^5\right)^{-1}$, we apply the power of a power rule, resulting in $z^{5 \cdot (-1)} = z^{-5}$.

Q: What is the rule for multiplying exponents with the same base?

A: When multiplying exponents with the same base, we add the exponents. For example, $x^3 \cdot x^4 = x^{3+4} = x^7$.

Q: How do we divide exponents with the same base?

A: When dividing exponents with the same base, we subtract the exponents. For example, $\frac{x^3}{x^4} = x^{3-4} = x^{-1}$.

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

A: When raising a power to a power, we multiply the exponents. For example, $(x^2)^3 = x^{2 \cdot 3} = x^6$.

Q: How do we simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, we can rewrite the expression with a positive exponent by moving the base to the other side of the fraction. For example, $\frac{1}{x^3} = x^{-3}$.

Example Problems

• Evaluate the expression $\left(a^4\right)^{-2}$.
• Simplify the expression $\left(b^{-3}\right)^{-1}$.
• Evaluate the expression $\left(c^2\right)^3$.

Solution

• To evaluate the expression $\left(a^4\right)^{-2}$, we apply the power of a power rule, resulting in $a^{4 \cdot (-2)} = a^{-8}$.
• To simplify the expression $\left(b^{-3}\right)^{-1}$, we apply the power of a power rule, resulting in $b^{(-3) \cdot (-1)} = b^3$.
• To evaluate the expression $\left(c^2\right)^3$, we apply the power of a power rule, resulting in $c^{2 \cdot 3} = c^6$.

Final Thoughts

In conclusion, evaluating expressions with exponents and powers requires a thorough understanding of the exponent rules. By applying these rules, we can simplify complex expressions and arrive at the final value. This Q&A article provides a comprehensive guide to evaluating expressions with exponents and powers, and we hope it has been helpful in addressing your questions and concerns.