Simplify The Expression:$\[ \left(\frac{\left(12 X Z^2\right)^{1 / 2}}{\left(3 Y^3 Z\right)^{1 / 2}}\right)^{-3} \\]

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In this article, we will focus on simplifying a specific expression involving exponents and fractions. We will break down the problem into manageable steps, and by the end of this guide, you will be able to simplify the given expression with ease.

Understanding the Expression

The given expression is:

$$\left(\frac{\left(12 x z^2\right)^{1 / 2}}{\left(3 y^3 z\right)^{1 / 2}}\right)^{-3}$$

This expression involves exponents, fractions, and variables. To simplify it, we need to understand the rules of exponents and how to manipulate fractions.

Rule 1: Exponent Rules

When dealing with exponents, there are several rules that we need to follow:

• Product Rule: When multiplying two numbers with the same base, we add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power Rule: When raising a power to another power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Quotient Rule: When dividing two numbers with the same base, we subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Rule 2: Fraction Manipulation

When dealing with fractions, there are several rules that we need to follow:

• Inverting and Multiplying: When we want to get rid of a fraction, we can invert the fraction and multiply. For example, $\frac{1}{a} \cdot a = 1$.
• Cancelling Common Factors: When we have a fraction with common factors in the numerator and denominator, we can cancel them out. For example, $\frac{2 \cdot 3}{2 \cdot 4} = \frac{3}{4}$.

Simplifying the Expression

Now that we have understood the rules of exponents and fraction manipulation, let's simplify the given expression.

Step 1: Apply the Quotient Rule

The given expression involves a fraction with exponents in the numerator and denominator. We can apply the quotient rule to simplify it:

$$\left(\frac{\left(12 x z^2\right)^{1 / 2}}{\left(3 y^3 z\right)^{1 / 2}}\right)^{-3} = \left(\frac{12 x z^2}{3 y^3 z}\right)^{-3 \cdot \frac{1}{2}}$$

Step 2: Simplify the Fraction

Now that we have applied the quotient rule, we can simplify the fraction:

$$\left(\frac{12 x z^2}{3 y^3 z}\right)^{-3 \cdot \frac{1}{2}} = \left(\frac{4 x z}{y^3}\right)^{-\frac{3}{2}}$$

Step 3: Apply the Power Rule

Now that we have simplified the fraction, we can apply the power rule to simplify the expression:

$$\left(\frac{4 x z}{y^3}\right)^{-\frac{3}{2}} = \frac{y^{\frac{9}{2}}}{4 x z^{\frac{3}{2}}}$$

Step 4: Simplify the Expression

Now that we have applied the power rule, we can simplify the expression:

$$\frac{y^{\frac{9}{2}}}{4 x z^{\frac{3}{2}}} = \frac{y^{\frac{9}{2}}}{4 x z^{\frac{3}{2}}}$$

Conclusion

In this article, we have simplified a complex expression involving exponents and fractions. We have applied the quotient rule, simplified the fraction, applied the power rule, and finally simplified the expression. By following these steps, you can simplify any expression involving exponents and fractions.

Final Answer

The final answer is $\boxed{\frac{y^{\frac{9}{2}}}{4 x z^{\frac{3}{2}}}}$.

Discussion

This problem is a great example of how to simplify complex expressions involving exponents and fractions. By applying the quotient rule, simplifying the fraction, applying the power rule, and finally simplifying the expression, we can arrive at the final answer. This problem requires a good understanding of algebraic manipulation and exponent rules.

Related Problems

If you want to practice more problems like this, here are some related problems:

• Simplify the expression: $\left(\frac{\left(2 x y^2\right)^{1 / 2}}{\left(3 z^3 w\right)^{1 / 2}}\right)^{-2}$
• Simplify the expression: $\left(\frac{\left(5 a b^2\right)^{1 / 2}}{\left(2 c^3 d\right)^{1 / 2}}\right)^{-1}$
• Simplify the expression: $\left(\frac{\left(7 e f^2\right)^{1 / 2}}{\left(3 g^3 h\right)^{1 / 2}}\right)^{-4}$

These problems require the same level of algebraic manipulation and exponent rules as the original problem. By practicing these problems, you can improve your skills in simplifying complex expressions involving exponents and fractions.

Introduction

In our previous article, we simplified a complex expression involving exponents and fractions. We applied the quotient rule, simplified the fraction, applied the power rule, and finally simplified the expression. In this article, we will provide a Q&A guide to algebraic manipulation, covering common questions and topics related to simplifying expressions.

Q&A Guide

Q: What is the quotient rule in algebraic manipulation?

A: The quotient rule states that when dividing two numbers with the same base, we subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Q: How do I simplify a fraction with exponents in the numerator and denominator?

A: To simplify a fraction with exponents in the numerator and denominator, we can apply the quotient rule. For example, $\left(\frac{\left(12 x z^2\right)^{1 / 2}}{\left(3 y^3 z\right)^{1 / 2}}\right)^{-3} = \left(\frac{12 x z^2}{3 y^3 z}\right)^{-3 \cdot \frac{1}{2}}$.

Q: What is the power rule in algebraic manipulation?

A: The power rule states that when raising a power to another power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.

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

A: To simplify an expression with multiple exponents, we can apply the power rule. For example, $(a^m)^n = a^{m \cdot n}$.

Q: What is the difference between a quotient and a product in algebraic manipulation?

A: A quotient is the result of dividing two numbers, while a product is the result of multiplying two numbers. For example, $\frac{a}{b}$ is a quotient, while $a \cdot b$ is a product.

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

A: To simplify an expression with a negative exponent, we can apply the rule that $a^{-n} = \frac{1}{a^n}$. For example, $a^{-3} = \frac{1}{a^3}$.

Q: What is the final answer to the original problem?

A: The final answer to the original problem is $\boxed{\frac{y^{\frac{9}{2}}}{4 x z^{\frac{3}{2}}}}$.

Common Mistakes

When simplifying expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Forgetting to apply the quotient rule: When simplifying a fraction with exponents in the numerator and denominator, don't forget to apply the quotient rule.
• Not simplifying the fraction: Make sure to simplify the fraction before applying the power rule.
• Not applying the power rule: When simplifying an expression with multiple exponents, don't forget to apply the power rule.
• Not checking for negative exponents: Make sure to check for negative exponents and apply the rule that $a^{-n} = \frac{1}{a^n}$.

Conclusion

In this article, we provided a Q&A guide to algebraic manipulation, covering common questions and topics related to simplifying expressions. We discussed the quotient rule, power rule, and how to simplify expressions with multiple exponents. We also covered common mistakes to avoid when simplifying expressions. By following these guidelines, you can improve your skills in simplifying complex expressions involving exponents and fractions.

Final Answer

The final answer is $\boxed{\frac{y^{\frac{9}{2}}}{4 x z^{\frac{3}{2}}}}$.

Discussion

This problem is a great example of how to simplify complex expressions involving exponents and fractions. By applying the quotient rule, simplifying the fraction, applying the power rule, and finally simplifying the expression, we can arrive at the final answer. This problem requires a good understanding of algebraic manipulation and exponent rules.

Related Problems

If you want to practice more problems like this, here are some related problems:

• Simplify the expression: $\left(\frac{\left(2 x y^2\right)^{1 / 2}}{\left(3 z^3 w\right)^{1 / 2}}\right)^{-2}$
• Simplify the expression: $\left(\frac{\left(5 a b^2\right)^{1 / 2}}{\left(2 c^3 d\right)^{1 / 2}}\right)^{-1}$
• Simplify the expression: $\left(\frac{\left(7 e f^2\right)^{1 / 2}}{\left(3 g^3 h\right)^{1 / 2}}\right)^{-4}$

These problems require the same level of algebraic manipulation and exponent rules as the original problem. By practicing these problems, you can improve your skills in simplifying complex expressions involving exponents and fractions.