Simplify The Expression:$\left(w^{-4} X^{-5} Y^2\right)^6 \times \frac{y^{[?]}}{w X}$

Introduction

Algebraic expressions can be complex and daunting, but with the right techniques and strategies, they can be simplified to reveal their underlying structure. In this article, we will explore the process of simplifying a given expression, which involves manipulating exponents, fractions, and variables. By the end of this guide, you will be equipped with the skills and knowledge to tackle even the most challenging algebraic expressions.

Understanding Exponents and Variables

Before we dive into the simplification process, it's essential to understand the basics of exponents and variables. Exponents are a shorthand way of representing repeated multiplication. For example, $x^3$ means $x \times x \times x$. Variables, on the other hand, are letters or symbols that represent unknown values. In the given expression, $w$, $x$, and $y$ are variables.

Simplifying the Expression

The given expression is $\left(w^{-4} x^{-5} y^2\right)^6 \times \frac{y^{[?]}}{w x}$. To simplify this expression, we need to apply the rules of exponents and fractions.

Rule 1: Exponentiation

When an exponent is raised to another exponent, we multiply the exponents. In this case, we have $\left(w^{-4} x^{-5} y^2\right)^6$. Using the rule of exponentiation, we can rewrite this as $w^{-24} x^{-30} y^{12}$.

Rule 2: Fractional Exponents

A fractional exponent can be rewritten as a radical. For example, $x^{\frac{1}{2}}$ means $\sqrt{x}$. However, in this case, we don't have any fractional exponents, so we can move on to the next rule.

Rule 3: Multiplication of Exponents

When we multiply two or more variables with the same base, we add their exponents. In this case, we have $w^{-24} x^{-30} y^{12} \times \frac{y^{[?]}}{w x}$. To simplify this expression, we need to multiply the exponents of $w$, $x$, and $y$.

Simplifying the Expression

Now that we have applied the rules of exponents, we can simplify the expression. We have $w^{-24} x^{-30} y^{12} \times \frac{y^{[?]}}{w x}$. To simplify this expression, we need to multiply the exponents of $w$, $x$, and $y$.

Step 1: Multiply the Exponents of $w$

When we multiply two or more variables with the same base, we add their exponents. In this case, we have $w^{-24} \times \frac{1}{w}$. To simplify this expression, we need to add the exponents of $w$. We get $w^{-24-1} = w^{-25}$.

Step 2: Multiply the Exponents of $x$

When we multiply two or more variables with the same base, we add their exponents. In this case, we have $x^{-30} \times \frac{1}{x}$. To simplify this expression, we need to add the exponents of $x$. We get $x^{-30-1} = x^{-31}$.

Step 3: Multiply the Exponents of $y$

When we multiply two or more variables with the same base, we add their exponents. In this case, we have y^{12} \times y^{[?]}}. To simplify this expression, we need to add the exponents of $y$. We get $y^{12+[?]}$.

Step 4: Simplify the Expression

Now that we have multiplied the exponents of $w$, $x$, and $y$, we can simplify the expression. We have $w^{-25} x^{-31} y^{12+[?]}$. To simplify this expression, we need to add the exponents of $y$. We get $y^{12+[?]} = y^{12+1} = y^{13}$.

Conclusion

In this article, we have simplified the given expression using the rules of exponents and fractions. We have applied the rules of exponentiation, fractional exponents, and multiplication of exponents to simplify the expression. By following these steps, we have arrived at the simplified expression: $w^{-25} x^{-31} y^{13}$.

Final Answer

The final answer is $\boxed{w^{-25} x^{-31} y^{13}}$.

Frequently Asked Questions

• What is the rule of exponentiation? The rule of exponentiation states that when an exponent is raised to another exponent, we multiply the exponents.
• What is the rule of fractional exponents? A fractional exponent can be rewritten as a radical. For example, $x^{\frac{1}{2}}$ means $\sqrt{x}$.
• What is the rule of multiplication of exponents? When we multiply two or more variables with the same base, we add their exponents.

References

• [1] Algebraic Expressions, Khan Academy
• [2] Exponents and Variables, Mathway
• [3] Simplifying Algebraic Expressions, Purplemath
  

Introduction

In our previous article, we explored the process of simplifying a given expression, which involves manipulating exponents, fractions, and variables. We applied the rules of exponentiation, fractional exponents, and multiplication of exponents to simplify the expression. In this article, we will answer some frequently asked questions related to simplifying algebraic expressions.

Q&A

Q1: What is the rule of exponentiation?

A1: The rule of exponentiation states that when an exponent is raised to another exponent, we multiply the exponents. For example, $\left(w^{-4} x^{-5} y^2\right)^6$ means $w^{-24} x^{-30} y^{12}$.

Q2: What is the rule of fractional exponents?

A2: A fractional exponent can be rewritten as a radical. For example, $x^{\frac{1}{2}}$ means $\sqrt{x}$. However, in the case of the given expression, we don't have any fractional exponents.

Q3: What is the rule of multiplication of exponents?

A3: When we multiply two or more variables with the same base, we add their exponents. For example, $w^{-24} x^{-30} y^{12} \times \frac{y^{[?]}}{w x}$ means $w^{-25} x^{-31} y^{13}$.

Q4: How do I simplify an expression with multiple variables?

A4: To simplify an expression with multiple variables, you need to apply the rules of exponentiation, fractional exponents, and multiplication of exponents. Start by simplifying each variable separately, and then combine the simplified expressions.

Q5: What is the final answer to the given expression?

A5: The final answer to the given expression is $\boxed{w^{-25} x^{-31} y^{13}}$.

Q6: Can I use a calculator to simplify algebraic expressions?

A6: While calculators can be helpful in simplifying algebraic expressions, it's always best to use the rules of exponentiation, fractional exponents, and multiplication of exponents to simplify expressions manually. This will help you understand the underlying structure of the expression and ensure that you arrive at the correct solution.

Q7: How do I apply the rules of exponentiation, fractional exponents, and multiplication of exponents?

A7: To apply the rules of exponentiation, fractional exponents, and multiplication of exponents, follow these steps:

1. Simplify each variable separately using the rules of exponentiation and fractional exponents.
2. Combine the simplified expressions using the rule of multiplication of exponents.
3. Simplify the resulting expression to arrive at the final answer.

Conclusion

In this article, we have answered some frequently asked questions related to simplifying algebraic expressions. We have applied the rules of exponentiation, fractional exponents, and multiplication of exponents to simplify the given expression. By following these steps, you can simplify any algebraic expression and arrive at the correct solution.

Final Answer

The final answer is $\boxed{w^{-25} x^{-31} y^{13}}$.

Frequently Asked Questions

• What is the rule of exponentiation?
• What is the rule of fractional exponents?
• What is the rule of multiplication of exponents?
• How do I simplify an expression with multiple variables?
• What is the final answer to the given expression?
• Can I use a calculator to simplify algebraic expressions?
• How do I apply the rules of exponentiation, fractional exponents, and multiplication of exponents?

References

• [1] Algebraic Expressions, Khan Academy
• [2] Exponents and Variables, Mathway
• [3] Simplifying Algebraic Expressions, Purplemath