Simplify The Expression:$\frac{6 X^{-4}}{4 X^2 Y^{-5}}$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it is essential to understand the rules and techniques involved in simplifying expressions. In this article, we will focus on simplifying the given expression $\frac{6 x^{-4}}{4 x^2 y^{-5}}$. We will use the properties of exponents and the rules of algebra to simplify the expression.

Understanding the Properties of Exponents

Before we can simplify the expression, we need to understand the properties of exponents. The properties of exponents state that:

• When we multiply two numbers with the same base, we add their exponents.
• When we divide two numbers with the same base, we subtract their exponents.
• When we raise a number with an exponent to another power, we multiply the exponents.

Simplifying the Expression

To simplify the expression $\frac{6 x^{-4}}{4 x^2 y^{-5}}$, we need to use the properties of exponents. We can start by simplifying the numerator and denominator separately.

Simplifying the Numerator

The numerator is $6 x^{-4}$. We can simplify this expression by using the property of exponents that states that when we raise a number with an exponent to another power, we multiply the exponents. In this case, we can rewrite the expression as $6x^{-4} = \frac{6}{x^4}$.

Simplifying the Denominator

The denominator is $4 x^2 y^{-5}$. We can simplify this expression by using the property of exponents that states that when we divide two numbers with the same base, we subtract their exponents. In this case, we can rewrite the expression as $4x^2y^{-5} = \frac{4y^{-5}}{x^{-2}}$.

Combining the Simplified Numerator and Denominator

Now that we have simplified the numerator and denominator, we can combine them to get the simplified expression. We can do this by multiplying the numerator and denominator by the reciprocal of the denominator.

$\frac{6}{x^4} \div \frac{4y^{-5}}{x^{-2}} = \frac{6}{x^4} \times \frac{x^{-2}}{4y^{-5}}$

Applying the Properties of Exponents

Now that we have combined the simplified numerator and denominator, we can apply the properties of exponents to simplify the expression further. We can do this by using the property of exponents that states that when we multiply two numbers with the same base, we add their exponents.

$\frac{6}{x^4} \times \frac{x^{-2}}{4y^{-5}} = \frac{6x^{-2}}{4x^4y^{-5}}$

Simplifying the Expression Further

Now that we have applied the properties of exponents, we can simplify the expression further by combining the exponents in the numerator and denominator.

$\frac{6x^{-2}}{4x^4y^{-5}} = \frac{6}{4x^6y^{-5}}$

Final Simplified Expression

The final simplified expression is $\frac{6}{4x^6y^{-5}}$. We can simplify this expression further by dividing the numerator and denominator by their greatest common divisor, which is 2.

$\frac{6}{4x^6y^{-5}} = \frac{3}{2x^6y^{-5}}$

Conclusion

In this article, we have simplified the expression $\frac{6 x^{-4}}{4 x^2 y^{-5}}$ using the properties of exponents and the rules of algebra. We have shown that the final simplified expression is $\frac{3}{2x^6y^{-5}}$. This expression is in its simplest form, and it cannot be simplified further.

Frequently Asked Questions

• What is the property of exponents that states when we multiply two numbers with the same base, we add their exponents? The property of exponents that states when we multiply two numbers with the same base, we add their exponents is: $a^m \times a^n = a^{m+n}$.
• What is the property of exponents that states when we divide two numbers with the same base, we subtract their exponents? The property of exponents that states when we divide two numbers with the same base, we subtract their exponents is: $\frac{a^m}{a^n} = a^{m-n}$.
• What is the property of exponents that states when we raise a number with an exponent to another power, we multiply the exponents? The property of exponents that states when we raise a number with an exponent to another power, we multiply the exponents is: $(a^m)^n = a^{mn}$.

References

• [1] "Algebra" by Michael Artin, 2nd edition, Prentice Hall, 2011.
• [2] "Calculus" by Michael Spivak, 4th edition, Publish or Perish, 2008.
• [3] "Mathematics for the Nonmathematician" by Morris Kline, 2nd edition, Dover Publications, 2007.

Further Reading

• "Simplifying Algebraic Expressions" by Math Open Reference, 2020.
• "Properties of Exponents" by Mathway, 2020.
• "Algebraic Expressions" by Khan Academy, 2020.
  

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Introduction

In our previous article, we simplified the expression $\frac{6 x^{-4}}{4 x^2 y^{-5}}$ using the properties of exponents and the rules of algebra. In this article, we will answer some frequently asked questions related to simplifying algebraic expressions.

Q&A

Q: What is the property of exponents that states when we multiply two numbers with the same base, we add their exponents?

A: The property of exponents that states when we multiply two numbers with the same base, we add their exponents is: $a^m \times a^n = a^{m+n}$.

Q: What is the property of exponents that states when we divide two numbers with the same base, we subtract their exponents?

A: The property of exponents that states when we divide two numbers with the same base, we subtract their exponents is: $\frac{a^m}{a^n} = a^{m-n}$.

Q: What is the property of exponents that states when we raise a number with an exponent to another power, we multiply the exponents?

A: The property of exponents that states when we raise a number with an exponent to another power, we multiply the exponents is: $(a^m)^n = a^{mn}$.

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

A: To simplify an expression with negative exponents, you can use the property of exponents that states that $a^{-m} = \frac{1}{a^m}$. For example, $x^{-4} = \frac{1}{x^4}$.

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

A: To simplify an expression with fractional exponents, you can use the property of exponents that states that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. For example, $x^{\frac{1}{2}} = \sqrt{x}$.

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change, while a constant is a value that does not change.

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

A: To simplify an expression with multiple variables, you can use the properties of exponents and the rules of algebra. For example, $\frac{6x^{-4}y^{-5}}{4x^2} = \frac{6}{4x^6y^{-5}}$.

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

A: The final simplified expression for the given problem is $\frac{3}{2x^6y^{-5}}$.

Conclusion

In this article, we have answered some frequently asked questions related to simplifying algebraic expressions. We have shown that the properties of exponents and the rules of algebra can be used to simplify expressions with negative exponents, fractional exponents, and multiple variables.

Frequently Asked Questions

• What is the property of exponents that states when we multiply two numbers with the same base, we add their exponents?
• What is the property of exponents that states when we divide two numbers with the same base, we subtract their exponents?
• What is the property of exponents that states when we raise a number with an exponent to another power, we multiply the exponents?
• How do I simplify an expression with negative exponents?
• How do I simplify an expression with fractional exponents?
• What is the difference between a variable and a constant?
• How do I simplify an expression with multiple variables?
• What is the final simplified expression for the given problem?

References

• [1] "Algebra" by Michael Artin, 2nd edition, Prentice Hall, 2011.
• [2] "Calculus" by Michael Spivak, 4th edition, Publish or Perish, 2008.
• [3] "Mathematics for the Nonmathematician" by Morris Kline, 2nd edition, Dover Publications, 2007.

Further Reading

• "Simplifying Algebraic Expressions" by Math Open Reference, 2020.
• "Properties of Exponents" by Mathway, 2020.
• "Algebraic Expressions" by Khan Academy, 2020.