Select The Correct Answer.Which Expression Is Equivalent To The Given Expression?$\frac{\left(a B^2\right)^3}{b^5}$A. $\frac{a^3}{b}$ B. $\frac{a^4}{b}$ C. $a^3 B$ D. $a^3$

Understanding Exponent Rules

When dealing with algebraic expressions, it's essential to understand the rules of exponents. Exponents are a shorthand way of representing repeated multiplication of a number. For example, $a^3$ means $a \times a \times a$. In this article, we'll focus on simplifying exponents in algebraic expressions, specifically the expression $\frac{\left(a b^2\right)^3}{b^5}$.

The Given Expression

The given expression is $\frac{\left(a b^2\right)^3}{b^5}$. To simplify this expression, we need to apply the rules of exponents. Let's break down the expression step by step.

Step 1: Apply the Power of a Product Rule

The power of a product rule states that for any numbers $a$ and $b$ and any integer $n$, $\left(ab\right)^n = a^n b^n$. We can apply this rule to the expression $\left(a b^2\right)^3$.

$\left(a b^2\right)^3 = a^3 \left(b^2\right)^3$

Step 2: Apply the Power of a Power Rule

The power of a power rule states that for any number $a$ and any integers $m$ and $n$, $\left(a^m\right)^n = a^{mn}$. We can apply this rule to the expression $\left(b^2\right)^3$.

$\left(b^2\right)^3 = b^{2 \times 3} = b^6$

Step 3: Simplify the Expression

Now that we have simplified the numerator, we can rewrite the original expression as:

$\frac{a^3 b^6}{b^5}$

Step 4: Apply the Quotient of Powers Rule

The quotient of powers rule states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $\frac{a^m}{a^n} = a^{m-n}$. We can apply this rule to the expression $\frac{b^6}{b^5}$.

$\frac{b^6}{b^5} = b^{6-5} = b^1 = b$

Step 5: Simplify the Final Expression

Now that we have simplified the expression, we can rewrite the original expression as:

$a^3 b$

Conclusion

In conclusion, the expression $\frac{\left(a b^2\right)^3}{b^5}$ is equivalent to $a^3 b$. This is achieved by applying the rules of exponents, specifically the power of a product rule, the power of a power rule, and the quotient of powers rule.

Answer

The correct answer is:

C. $a^3 b$

Discussion

This problem requires a good understanding of the rules of exponents. The power of a product rule, the power of a power rule, and the quotient of powers rule are essential in simplifying algebraic expressions. By applying these rules, we can simplify complex expressions and arrive at the correct solution.

Additional Examples

Here are some additional examples of simplifying exponents in algebraic expressions:

• $\frac{\left(a^2 b\right)^2}{a^3} = \frac{a^4 b^2}{a^3} = a^{4-3} b^2 = a b^2$
• $\frac{\left(a b^2\right)^3}{a^2} = \frac{a^3 b^6}{a^2} = a^{3-2} b^6 = a b^6$
• $\frac{\left(a^2 b\right)^2}{b^3} = \frac{a^4 b^2}{b^3} = a^4 b^{2-3} = a^4 b^{-1} = \frac{a^4}{b}$

Understanding Exponents in Algebra

Exponents are a fundamental concept in algebra, and understanding how to simplify them is crucial for solving complex equations and expressions. In this article, we'll provide a Q&A section to help you better understand the rules of exponents and how to apply them to simplify algebraic expressions.

Q: What is the power of a product rule?

A: The power of a product rule states that for any numbers $a$ and $b$ and any integer $n$, $\left(ab\right)^n = a^n b^n$. This rule allows us to simplify expressions by distributing the exponent to each factor.

Q: How do I apply the power of a product rule?

A: To apply the power of a product rule, simply distribute the exponent to each factor. For example, $\left(a b^2\right)^3 = a^3 \left(b^2\right)^3 = a^3 b^6$.

Q: What is the power of a power rule?

A: The power of a power rule states that for any number $a$ and any integers $m$ and $n$, $\left(a^m\right)^n = a^{mn}$. This rule allows us to simplify expressions by combining exponents.

Q: How do I apply the power of a power rule?

A: To apply the power of a power rule, simply multiply the exponents. For example, $\left(b^2\right)^3 = b^{2 \times 3} = b^6$.

Q: What is the quotient of powers rule?

A: The quotient of powers rule states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $\frac{a^m}{a^n} = a^{m-n}$. This rule allows us to simplify expressions by subtracting exponents.

Q: How do I apply the quotient of powers rule?

A: To apply the quotient of powers rule, simply subtract the exponents. For example, $\frac{b^6}{b^5} = b^{6-5} = b^1 = b$.

Q: How do I simplify the expression $\frac{\left(a b^2\right)^3}{b^5}$?

A: To simplify the expression $\frac{\left(a b^2\right)^3}{b^5}$, we can apply the rules of exponents as follows:

1. Apply the power of a product rule: $\left(a b^2\right)^3 = a^3 \left(b^2\right)^3 = a^3 b^6$
2. Apply the power of a power rule: $\left(b^2\right)^3 = b^{2 \times 3} = b^6$
3. Simplify the expression: $\frac{a^3 b^6}{b^5} = a^3 b^{6-5} = a^3 b^1 = a^3 b$

Q: What is the final simplified expression?

A: The final simplified expression is $a^3 b$.

Conclusion

In conclusion, understanding the rules of exponents is crucial for simplifying algebraic expressions. By applying the power of a product rule, the power of a power rule, and the quotient of powers rule, we can simplify complex expressions and arrive at the correct solution. We hope this Q&A section has helped you better understand the rules of exponents and how to apply them to simplify algebraic expressions.

Additional Resources

For more information on simplifying exponents in algebraic expressions, we recommend the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram Alpha: Exponents and Exponential Functions

These resources provide a comprehensive overview of exponents and exponential functions, including examples, exercises, and interactive tools to help you practice and reinforce your understanding.