Simplify: ( 2 X − 2 8 X 5 Y − 1 ) − 2 \left(\frac{2 X^{-2}}{8 X^5 Y^{-1}}\right)^{-2} ( 8 X 5 Y − 1 2 X − 2 ​ ) − 2

Introduction

Simplifying complex mathematical expressions is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. In this article, we will focus on simplifying the expression $\left(\frac{2 x^{-2}}{8 x^5 y^{-1}}\right)^{-2}$, which involves exponents, fractions, and variables. We will break down the expression step by step, using the rules of exponents and fraction simplification to arrive at the final simplified form.

Understanding Exponents and Fractions

Before we dive into simplifying the expression, let's review the rules of exponents and fractions. Exponents are used to represent repeated multiplication of a number or variable. For example, $x^2$ represents $x$ multiplied by itself twice, or $x \cdot x \cdot x$. When we have a negative exponent, it means we are taking the reciprocal of the expression. For example, $x^{-2}$ represents $\frac{1}{x^2}$.

Fractions are used to represent a part of a whole. In the expression $\frac{2 x^{-2}}{8 x^5 y^{-1}}$, the numerator is $2 x^{-2}$ and the denominator is $8 x^5 y^{-1}$. To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD.

Simplifying the Expression

Now that we have reviewed the rules of exponents and fractions, let's simplify the expression $\left(\frac{2 x^{-2}}{8 x^5 y^{-1}}\right)^{-2}$. To simplify this expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the exponent in the numerator: $x^{-2}$ represents $\frac{1}{x^2}$.
2. Evaluate the exponent in the denominator: $x^5$ remains the same.
3. Simplify the fraction: $\frac{2}{8}$ can be simplified by dividing both the numerator and denominator by 2, resulting in $\frac{1}{4}$.
4. Simplify the expression inside the parentheses: $\frac{\frac{1}{x^2}}{x^5 y^{-1}}$ can be simplified by multiplying the numerator and denominator by $x^2$, resulting in $\frac{1}{x^7 y^{-1}}$.
5. Simplify the exponent outside the parentheses: $-2$ represents taking the reciprocal of the expression, so we need to take the reciprocal of $\frac{1}{x^7 y^{-1}}$, resulting in $x^{14} y$.

Final Simplified Form

After following the order of operations and simplifying the expression step by step, we arrive at the final simplified form: $x^{14} y$.

Conclusion

Simplifying complex mathematical expressions requires a deep understanding of the underlying concepts, including exponents and fractions. By following the order of operations and simplifying the expression step by step, we can arrive at the final simplified form. In this article, we simplified the expression $\left(\frac{2 x^{-2}}{8 x^5 y^{-1}}\right)^{-2}$, which involved exponents, fractions, and variables. We hope this article has provided valuable insights and skills for simplifying complex mathematical expressions.

Additional Tips and Tricks

• When simplifying expressions with exponents, make sure to follow the order of operations (PEMDAS).
• When simplifying fractions, find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD.
• When simplifying expressions with variables, make sure to use the correct rules for exponents and fractions.
• Practice simplifying complex mathematical expressions to develop your skills and confidence.

Frequently Asked Questions

• Q: What is the rule for simplifying expressions with exponents? A: The rule for simplifying expressions with exponents is to follow the order of operations (PEMDAS) and simplify the expression step by step.
• Q: How do I simplify a fraction? A: To simplify a fraction, find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD.
• Q: What is the final simplified form of the expression $\left(\frac{2 x^{-2}}{8 x^5 y^{-1}}\right)^{-2}$? A: The final simplified form of the expression $\left(\frac{2 x^{-2}}{8 x^5 y^{-1}}\right)^{-2}$ is $x^{14} y$.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for general information and are not specific to the topic of simplifying the expression $\left(\frac{2 x^{-2}}{8 x^5 y^{-1}}\right)^{-2}$.

Introduction

In our previous article, we simplified the expression $\left(\frac{2 x^{-2}}{8 x^5 y^{-1}}\right)^{-2}$, which involved exponents, fractions, and variables. We received many questions from readers who were struggling to understand the concept of simplifying complex mathematical expressions. In this article, we will answer some of the most frequently asked questions (FAQs) related to simplifying expressions with exponents and fractions.

Q&A

Q: What is the rule for simplifying expressions with exponents?

A: The rule for simplifying expressions with exponents is to follow the order of operations (PEMDAS) and simplify the expression step by step. When simplifying expressions with exponents, make sure to use the correct rules for exponents and fractions.

Q: How do I simplify a fraction?

A: To simplify a fraction, find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD. For example, to simplify the fraction $\frac{2 x^{-2}}{8 x^5 y^{-1}}$, we can divide both the numerator and denominator by 2, resulting in $\frac{1}{4 x^5 y^{-1}}$.

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

A: A positive exponent represents repeated multiplication of a number or variable, while a negative exponent represents taking the reciprocal of the expression. For example, $x^2$ represents $x$ multiplied by itself twice, while $x^{-2}$ represents $\frac{1}{x^2}$.

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

A: To simplify an expression with multiple variables, make sure to use the correct rules for exponents and fractions. For example, to simplify the expression $\left(\frac{2 x^{-2} y^3}{8 x^5 y^{-1}}\right)^{-2}$, we can follow the order of operations (PEMDAS) and simplify the expression step by step.

Q: What is the final simplified form of the expression $\left(\frac{2 x^{-2}}{8 x^5 y^{-1}}\right)^{-2}$?

A: The final simplified form of the expression $\left(\frac{2 x^{-2}}{8 x^5 y^{-1}}\right)^{-2}$ is $x^{14} y$.

Q: How do I know when to use a positive or negative exponent?

A: When simplifying expressions with exponents, make sure to use the correct rules for exponents and fractions. If the exponent is positive, it represents repeated multiplication of a number or variable. If the exponent is negative, it represents taking the reciprocal of the expression.

Q: Can I simplify an expression with a variable in the denominator?

A: Yes, you can simplify an expression with a variable in the denominator. To simplify an expression with a variable in the denominator, make sure to use the correct rules for exponents and fractions. For example, to simplify the expression $\frac{2 x^{-2}}{x^5 y^{-1}}$, we can follow the order of operations (PEMDAS) and simplify the expression step by step.

Conclusion

Simplifying complex mathematical expressions requires a deep understanding of the underlying concepts, including exponents and fractions. By following the order of operations (PEMDAS) and simplifying the expression step by step, we can arrive at the final simplified form. In this article, we answered some of the most frequently asked questions (FAQs) related to simplifying expressions with exponents and fractions. We hope this article has provided valuable insights and skills for simplifying complex mathematical expressions.

Additional Tips and Tricks

• When simplifying expressions with exponents, make sure to follow the order of operations (PEMDAS).
• When simplifying fractions, find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD.
• When simplifying expressions with variables, make sure to use the correct rules for exponents and fractions.
• Practice simplifying complex mathematical expressions to develop your skills and confidence.

Frequently Asked Questions

• Q: What is the rule for simplifying expressions with exponents? A: The rule for simplifying expressions with exponents is to follow the order of operations (PEMDAS) and simplify the expression step by step.
• Q: How do I simplify a fraction? A: To simplify a fraction, find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD.
• Q: What is the final simplified form of the expression $\left(\frac{2 x^{-2}}{8 x^5 y^{-1}}\right)^{-2}$? A: The final simplified form of the expression $\left(\frac{2 x^{-2}}{8 x^5 y^{-1}}\right)^{-2}$ is $x^{14} y$.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for general information and are not specific to the topic of simplifying the expression $\left(\frac{2 x^{-2}}{8 x^5 y^{-1}}\right)^{-2}$.