Simplify The Expression Completely, And Use Only Positive Exponents: ${ \frac{x {-6}}{y {-8}} \times \frac{y 8}{x 6} }$

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Introduction

When dealing with algebraic expressions, simplifying them is a crucial step in solving mathematical problems. In this article, we will focus on simplifying the given expression, which involves negative exponents and fractions. We will use the properties of exponents and fractions to simplify the expression completely, resulting in only positive exponents.

The Given Expression

The given expression is:

x−6y−8×y8x6\frac{x^{-6}}{y^{-8}} \times \frac{y^8}{x^6}

This expression involves negative exponents and fractions, making it a bit challenging to simplify. However, with the right approach, we can simplify it completely and obtain an expression with only positive exponents.

Properties of Exponents

Before we dive into simplifying the expression, let's review some important properties of exponents:

  • Product of Powers: When multiplying two powers with the same base, we add the exponents. For example, xa×xb=xa+bx^a \times x^b = x^{a+b}.
  • Quotient of Powers: When dividing two powers with the same base, we subtract the exponents. For example, xaxb=xa−b\frac{x^a}{x^b} = x^{a-b}.
  • Power of a Power: When raising a power to another power, we multiply the exponents. For example, (xa)b=xab(x^a)^b = x^{ab}.

Simplifying the Expression

Now that we have reviewed the properties of exponents, let's simplify the given expression:

x−6y−8×y8x6\frac{x^{-6}}{y^{-8}} \times \frac{y^8}{x^6}

To simplify this expression, we can start by using the quotient of powers property to simplify the first fraction:

x−6y−8=y8x6\frac{x^{-6}}{y^{-8}} = \frac{y^8}{x^6}

Now, we can multiply the two fractions together:

y8x6×y8x6=y16x12\frac{y^8}{x^6} \times \frac{y^8}{x^6} = \frac{y^{16}}{x^{12}}

Using the Product of Powers Property

Now that we have simplified the expression, let's use the product of powers property to simplify it further:

y16x12=y16x12×x12x12=y16×x12x24\frac{y^{16}}{x^{12}} = \frac{y^{16}}{x^{12}} \times \frac{x^{12}}{x^{12}} = \frac{y^{16} \times x^{12}}{x^{24}}

Simplifying the Expression Further

Now that we have used the product of powers property, let's simplify the expression further:

y16×x12x24=y16x12\frac{y^{16} \times x^{12}}{x^{24}} = \frac{y^{16}}{x^{12}}

Conclusion

In this article, we have simplified the given expression, which involved negative exponents and fractions. We used the properties of exponents and fractions to simplify the expression completely, resulting in only positive exponents. The final simplified expression is:

y16x12\frac{y^{16}}{x^{12}}

Final Answer

The final answer is y16x12\boxed{\frac{y^{16}}{x^{12}}}.

Frequently Asked Questions

Q: What is the property of exponents used to simplify the expression?

A: The product of powers property is used to simplify the expression.

Q: What is the final simplified expression?

A: The final simplified expression is y16x12\frac{y^{16}}{x^{12}}.

Q: What is the property of exponents used to simplify the expression further?

A: The product of powers property is used to simplify the expression further.

Step-by-Step Solution

Step 1: Simplify the first fraction using the quotient of powers property

x−6y−8=y8x6\frac{x^{-6}}{y^{-8}} = \frac{y^8}{x^6}

Step 2: Multiply the two fractions together

y8x6×y8x6=y16x12\frac{y^8}{x^6} \times \frac{y^8}{x^6} = \frac{y^{16}}{x^{12}}

Step 3: Use the product of powers property to simplify the expression further

y16x12=y16×x12x24\frac{y^{16}}{x^{12}} = \frac{y^{16} \times x^{12}}{x^{24}}

Step 4: Simplify the expression further

y16×x12x24=y16x12\frac{y^{16} \times x^{12}}{x^{24}} = \frac{y^{16}}{x^{12}}

Common Mistakes

Mistake 1: Not using the quotient of powers property to simplify the first fraction

x−6y−8≠x6y8\frac{x^{-6}}{y^{-8}} \neq \frac{x^6}{y^8}

Mistake 2: Not using the product of powers property to simplify the expression further

y16x12≠y16×x12x12\frac{y^{16}}{x^{12}} \neq \frac{y^{16} \times x^{12}}{x^{12}}

Tips and Tricks

Tip 1: Use the quotient of powers property to simplify fractions with negative exponents

x−6y−8=y8x6\frac{x^{-6}}{y^{-8}} = \frac{y^8}{x^6}

Tip 2: Use the product of powers property to simplify expressions with multiple powers

y16x12=y16×x12x24\frac{y^{16}}{x^{12}} = \frac{y^{16} \times x^{12}}{x^{24}}

Real-World Applications

Application 1: Simplifying algebraic expressions in mathematics

Simplifying algebraic expressions is a crucial step in solving mathematical problems. By using the properties of exponents and fractions, we can simplify expressions and obtain a final answer.

Application 2: Simplifying expressions in physics and engineering

Simplifying expressions is also important in physics and engineering, where complex mathematical expressions are often used to describe physical systems. By simplifying these expressions, we can obtain a better understanding of the underlying physics and make more accurate predictions.

Conclusion

In conclusion, simplifying the given expression involved using the properties of exponents and fractions. By following the steps outlined in this article, we were able to simplify the expression completely and obtain a final answer. The final simplified expression is y16x12\frac{y^{16}}{x^{12}}.

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Introduction

In our previous article, we simplified the given expression, which involved negative exponents and fractions. We used the properties of exponents and fractions to simplify the expression completely, resulting in only positive exponents. In this article, we will answer some frequently asked questions related to simplifying expressions with negative exponents and fractions.

Q&A

Q: What is the property of exponents used to simplify the expression?

A: The product of powers property is used to simplify the expression.

Q: What is the final simplified expression?

A: The final simplified expression is y16x12\frac{y^{16}}{x^{12}}.

Q: What is the property of exponents used to simplify the expression further?

A: The product of powers property is used to simplify the expression further.

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

A: To simplify an expression with a negative exponent, you can use the quotient of powers property. For example, x−6y−8=y8x6\frac{x^{-6}}{y^{-8}} = \frac{y^8}{x^6}.

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

A: To simplify an expression with multiple powers, you can use the product of powers property. For example, y16x12=y16×x12x24\frac{y^{16}}{x^{12}} = \frac{y^{16} \times x^{12}}{x^{24}}.

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 power and then taken as a reciprocal.

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

A: To simplify an expression with a fraction and a negative exponent, you can use the quotient of powers property and the product of powers property. For example, x−6y−8×y8x6=y16x12\frac{x^{-6}}{y^{-8}} \times \frac{y^8}{x^6} = \frac{y^{16}}{x^{12}}.

Common Mistakes

Mistake 1: Not using the quotient of powers property to simplify the first fraction

x−6y−8≠x6y8\frac{x^{-6}}{y^{-8}} \neq \frac{x^6}{y^8}

Mistake 2: Not using the product of powers property to simplify the expression further

y16x12≠y16×x12x12\frac{y^{16}}{x^{12}} \neq \frac{y^{16} \times x^{12}}{x^{12}}

Tips and Tricks

Tip 1: Use the quotient of powers property to simplify fractions with negative exponents

x−6y−8=y8x6\frac{x^{-6}}{y^{-8}} = \frac{y^8}{x^6}

Tip 2: Use the product of powers property to simplify expressions with multiple powers

y16x12=y16×x12x24\frac{y^{16}}{x^{12}} = \frac{y^{16} \times x^{12}}{x^{24}}

Real-World Applications

Application 1: Simplifying algebraic expressions in mathematics

Simplifying algebraic expressions is a crucial step in solving mathematical problems. By using the properties of exponents and fractions, we can simplify expressions and obtain a final answer.

Application 2: Simplifying expressions in physics and engineering

Simplifying expressions is also important in physics and engineering, where complex mathematical expressions are often used to describe physical systems. By simplifying these expressions, we can obtain a better understanding of the underlying physics and make more accurate predictions.

Conclusion

In conclusion, simplifying the given expression involved using the properties of exponents and fractions. By following the steps outlined in this article, we were able to simplify the expression completely and obtain a final answer. The final simplified expression is y16x12\frac{y^{16}}{x^{12}}. We hope that this article has provided you with a better understanding of how to simplify expressions with negative exponents and fractions.

Final Answer

The final answer is y16x12\boxed{\frac{y^{16}}{x^{12}}}.