Simplify The Expression:${ \frac{\left(-y {-2}\right) 3}{-2 X^3} }$

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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 focus on simplifying a specific expression involving negative exponents and fractions. We will break down the expression into manageable parts, apply the rules of exponents, and manipulate the terms to arrive at a simplified form.

The Expression to Simplify

The given expression is:

(−y−2)3−2x3\frac{\left(-y^{-2}\right)^3}{-2 x^3}

This expression involves a negative exponent, a fraction, and a cube of a binomial. Our goal is to simplify this expression by applying the rules of exponents and algebraic manipulation.

Step 1: Apply the Power Rule for Exponents

The power rule for exponents states that for any non-zero number aa and integers mm and nn, we have:

(am)n=am⋅n(a^m)^n = a^{m \cdot n}

We can apply this rule to the expression (−y−2)3\left(-y^{-2}\right)^3 to simplify it:

(−y−2)3=(−1)3⋅(y−2)3=−y−6\left(-y^{-2}\right)^3 = (-1)^3 \cdot (y^{-2})^3 = -y^{-6}

Step 2: Simplify the Fraction

Now that we have simplified the numerator, we can focus on the denominator. The denominator is −2x3-2 x^3. We can rewrite this as:

−2x3=−2⋅x3-2 x^3 = -2 \cdot x^3

Step 3: Combine the Numerator and Denominator

Now that we have simplified the numerator and denominator, we can combine them to form the simplified expression:

−y−6−2x3\frac{-y^{-6}}{-2 x^3}

Step 4: Apply the Quotient Rule for Exponents

The quotient rule for exponents states that for any non-zero numbers aa and bb and integers mm and nn, we have:

aman=am−n\frac{a^m}{a^n} = a^{m-n}

We can apply this rule to the expression −y−6−2x3\frac{-y^{-6}}{-2 x^3} to simplify it:

−y−6−2x3=1−2⋅1x3⋅y−6=−12x3⋅y−6\frac{-y^{-6}}{-2 x^3} = \frac{1}{-2} \cdot \frac{1}{x^3} \cdot y^{-6} = -\frac{1}{2 x^3} \cdot y^{-6}

Step 5: Simplify the Expression Further

We can simplify the expression further by applying the rule for negative exponents:

−12x3⋅y−6=−12x3⋅1y6=−12x3y6-\frac{1}{2 x^3} \cdot y^{-6} = -\frac{1}{2 x^3} \cdot \frac{1}{y^6} = -\frac{1}{2 x^3 y^6}

Conclusion

In this article, we simplified the expression (−y−2)3−2x3\frac{\left(-y^{-2}\right)^3}{-2 x^3} by applying the rules of exponents and algebraic manipulation. We broke down the expression into manageable parts, applied the power rule for exponents, simplified the fraction, combined the numerator and denominator, applied the quotient rule for exponents, and simplified the expression further. The final simplified expression is:

−12x3y6-\frac{1}{2 x^3 y^6}

This expression reveals the underlying structure of the original expression and provides a clear and concise representation of the relationship between the variables.

Frequently Asked Questions

  • Q: What is the power rule for exponents? A: The power rule for exponents states that for any non-zero number aa and integers mm and nn, we have: (am)n=am⋅n(a^m)^n = a^{m \cdot n}.
  • Q: What is the quotient rule for exponents? A: The quotient rule for exponents states that for any non-zero numbers aa and bb and integers mm and nn, we have: aman=am−n\frac{a^m}{a^n} = a^{m-n}.
  • Q: How do I simplify an expression involving negative exponents? A: To simplify an expression involving negative exponents, you can apply the rule for negative exponents, which states that for any non-zero number aa and integer nn, we have: a−n=1ana^{-n} = \frac{1}{a^n}.

Final Thoughts

Simplifying algebraic expressions is an essential skill in mathematics, and it requires a deep understanding of the rules of exponents and algebraic manipulation. By breaking down complex expressions into manageable parts and applying the rules of exponents, we can simplify even the most daunting expressions and reveal their underlying structure. Whether you are a student, a teacher, or a professional mathematician, mastering the art of simplifying algebraic expressions is essential for success in mathematics and beyond.

Introduction

In our previous article, we simplified the expression (−y−2)3−2x3\frac{\left(-y^{-2}\right)^3}{-2 x^3} by applying the rules of exponents and algebraic manipulation. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used in simplifying algebraic expressions.

Q&A Guide

Q: What is the power rule for exponents?

A: The power rule for exponents states that for any non-zero number aa and integers mm and nn, we have:

(am)n=am⋅n(a^m)^n = a^{m \cdot n}

This rule allows us to simplify expressions involving exponents by multiplying the exponents.

Q: What is the quotient rule for exponents?

A: The quotient rule for exponents states that for any non-zero numbers aa and bb and integers mm and nn, we have:

aman=am−n\frac{a^m}{a^n} = a^{m-n}

This rule allows us to simplify expressions involving fractions by subtracting the exponents.

Q: How do I simplify an expression involving negative exponents?

A: To simplify an expression involving negative exponents, you can apply the rule for negative exponents, which states that for any non-zero number aa and integer nn, we have:

a−n=1ana^{-n} = \frac{1}{a^n}

This rule allows us to rewrite negative exponents as fractions.

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

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the base is raised to a reciprocal power. For example, a3a^3 indicates that aa is raised to the power of 3, while a−3a^{-3} indicates that aa is raised to the reciprocal power of 3.

Q: How do I simplify an expression involving a fraction and an exponent?

A: To simplify an expression involving a fraction and an exponent, you can apply the quotient rule for exponents and the rule for negative exponents. For example, if you have the expression aman\frac{a^m}{a^n}, you can simplify it by subtracting the exponents and rewriting the negative exponent as a fraction.

Q: What is the final simplified expression for the given problem?

A: The final simplified expression for the given problem is:

−12x3y6-\frac{1}{2 x^3 y^6}

This expression reveals the underlying structure of the original expression and provides a clear and concise representation of the relationship between the variables.

Common Mistakes to Avoid

  • Not applying the power rule for exponents: Failing to apply the power rule for exponents can lead to incorrect simplifications.
  • Not applying the quotient rule for exponents: Failing to apply the quotient rule for exponents can lead to incorrect simplifications.
  • Not rewriting negative exponents as fractions: Failing to rewrite negative exponents as fractions can lead to incorrect simplifications.
  • Not simplifying expressions involving fractions and exponents: Failing to simplify expressions involving fractions and exponents can lead to incorrect simplifications.

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics, and it requires a deep understanding of the rules of exponents and algebraic manipulation. By applying the power rule for exponents, the quotient rule for exponents, and the rule for negative exponents, you can simplify even the most daunting expressions and reveal their underlying structure. Whether you are a student, a teacher, or a professional mathematician, mastering the art of simplifying algebraic expressions is essential for success in mathematics and beyond.

Final Thoughts

Simplifying algebraic expressions is a skill that requires practice and patience. By working through examples and applying the rules of exponents and algebraic manipulation, you can develop your skills and become proficient in simplifying even the most complex expressions. Remember to always apply the power rule for exponents, the quotient rule for exponents, and the rule for negative exponents to simplify expressions involving exponents and fractions. With practice and dedication, you can master the art of simplifying algebraic expressions and achieve success in mathematics and beyond.