Simplify The Expression: ${ -\frac{45^{x+1} \cdot 9 {x-2}}{9 {2x} \cdot 5^x} }$

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that mathematicians and scientists use to solve complex problems. In this article, we will delve into the world of algebraic manipulation and provide a step-by-step guide on how to simplify the given expression: ${ -\frac{45^{x+1} \cdot 9^{x-2}}{9{2x} \cdot 5^x} }$. By the end of this article, you will have a thorough understanding of how to simplify complex expressions and be able to apply this skill to various mathematical problems.

Understanding the Expression

Before we begin simplifying the expression, let's break it down and understand its components. The expression is a fraction, and it consists of two parts: the numerator and the denominator. The numerator is $45^{x+1} \cdot 9^{x-2}$, and the denominator is $9^{2x} \cdot 5^x$. To simplify this expression, we need to manipulate the numerator and the denominator separately and then combine them.

Simplifying the Numerator

The numerator is $45^{x+1} \cdot 9^{x-2}$. To simplify this expression, we can use the properties of exponents. We know that $45 = 9 \cdot 5$, so we can rewrite the numerator as $(9 \cdot 5)^{x+1} \cdot 9^{x-2}$. Using the property of exponents that states $(ab)^c = a^c \cdot b^c$, we can rewrite the numerator as $9^{x+1} \cdot 5^{x+1} \cdot 9^{x-2}$.

Simplifying the Denominator

The denominator is $9^{2x} \cdot 5^x$. To simplify this expression, we can use the properties of exponents. We know that $9 = 3^2$, so we can rewrite the denominator as $(3^2)^{2x} \cdot 5^x$. Using the property of exponents that states $(a^b)^c = a^{bc}$, we can rewrite the denominator as $3^{4x} \cdot 5^x$.

Combining the Numerator and the Denominator

Now that we have simplified the numerator and the denominator, we can combine them to get the final expression. We have $9^{x+1} \cdot 5^{x+1} \cdot 9^{x-2}$ in the numerator and $3^{4x} \cdot 5^x$ in the denominator. To combine these expressions, we can use the properties of exponents. We know that when we multiply two numbers with the same base, we can add their exponents. So, we can rewrite the numerator as $9^{x+1+x-2} \cdot 5^{x+1}$, which simplifies to $9^{2x-1} \cdot 5^{x+1}$.

Simplifying the Final Expression

Now that we have combined the numerator and the denominator, we can simplify the final expression. We have $9^{2x-1} \cdot 5^{x+1}$ in the numerator and $3^{4x} \cdot 5^x$ in the denominator. To simplify this expression, we can use the properties of exponents. We know that when we divide two numbers with the same base, we can subtract their exponents. So, we can rewrite the expression as $\frac{9^{2x-1} \cdot 5^{x+1}}{3^{4x} \cdot 5^x}$.

Canceling Out Common Factors

Now that we have simplified the final expression, we can cancel out common factors. We have $9^{2x-1} \cdot 5^{x+1}$ in the numerator and $3^{4x} \cdot 5^x$ in the denominator. We can cancel out the common factor of $5^x$ in the numerator and the denominator, which leaves us with $\frac{9^{2x-1} \cdot 5}{3^{4x}}$.

Simplifying the Final Expression Further

Now that we have canceled out the common factor, we can simplify the final expression further. We have $\frac{9^{2x-1} \cdot 5}{3^{4x}}$. To simplify this expression, we can use the properties of exponents. We know that $9 = 3^2$, so we can rewrite the expression as $\frac{(3^2)^{2x-1} \cdot 5}{3^{4x}}$. Using the property of exponents that states $(a^b)^c = a^{bc}$, we can rewrite the expression as $\frac{3^{4x-2} \cdot 5}{3^{4x}}$.

Canceling Out Common Factors Again

Now that we have simplified the final expression further, we can cancel out common factors again. We have $\frac{3^{4x-2} \cdot 5}{3^{4x}}$. We can cancel out the common factor of $3^{4x}$ in the numerator and the denominator, which leaves us with $3^{-2} \cdot 5$.

Simplifying the Final Expression Once More

Now that we have canceled out the common factor again, we can simplify the final expression once more. We have $3^{-2} \cdot 5$. To simplify this expression, we can use the properties of exponents. We know that $3^{-2} = \frac{1}{3^2}$, so we can rewrite the expression as $\frac{1}{3^2} \cdot 5$. Using the property of exponents that states $a^b \cdot a^c = a^{b+c}$, we can rewrite the expression as $\frac{5}{3^2}$.

Conclusion

In this article, we have simplified the expression ${ -\frac{45^{x+1} \cdot 9^{x-2}}{9{2x} \cdot 5^x} }$. We have broken down the expression into its components, simplified the numerator and the denominator separately, combined them, and canceled out common factors. By the end of this article, you should have a thorough understanding of how to simplify complex expressions and be able to apply this skill to various mathematical problems.

Final Answer

The final answer is $\boxed{\frac{5}{9}}$.

Frequently Asked Questions

• What is the process of simplifying an expression? The process of simplifying an expression involves breaking it down into its components, simplifying each component separately, combining them, and canceling out common factors.
• What are the properties of exponents that we used to simplify the expression? We used the properties of exponents that state $(ab)^c = a^c \cdot b^c$, $(a^b)^c = a^{bc}$, and $a^b \cdot a^c = a^{b+c}$ to simplify the expression.
• What is the final answer to the expression? The final answer to the expression is $\boxed{\frac{5}{9}}$.

References

• [1] Algebraic Manipulation, Wikipedia
• [2] Exponents, Khan Academy
• [3] Simplifying Expressions, Mathway

Further Reading

• Algebraic Manipulation, MIT OpenCourseWare
• Exponents, Wolfram MathWorld
• Simplifying Expressions, Purplemath
  

Introduction

In our previous article, we provided a step-by-step guide on how to simplify the expression ${ -\frac{45^{x+1} \cdot 9^{x-2}}{9{2x} \cdot 5^x} }$. We broke down the expression into its components, simplified the numerator and the denominator separately, combined them, and canceled out common factors. In this article, we will answer some of the most frequently asked questions about simplifying expressions and provide additional resources for further learning.

Q&A

Q: What is the process of simplifying an expression?

A: The process of simplifying an expression involves breaking it down into its components, simplifying each component separately, combining them, and canceling out common factors.

Q: What are the properties of exponents that we used to simplify the expression?

A: We used the properties of exponents that state $(ab)^c = a^c \cdot b^c$, $(a^b)^c = a^{bc}$, and $a^b \cdot a^c = a^{b+c}$ to simplify the expression.

Q: What is the final answer to the expression?

A: The final answer to the expression is $\boxed{\frac{5}{9}}$.

Q: How do I simplify a complex expression?

A: To simplify a complex expression, you need to break it down into its components, simplify each component separately, combine them, and cancel out common factors. You can use the properties of exponents to simplify the expression.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Not breaking down the expression into its components
• Not simplifying each component separately
• Not combining the components correctly
• Not canceling out common factors

Q: How do I know when to use the properties of exponents?

A: You should use the properties of exponents when you have an expression that involves exponents and you need to simplify it. The properties of exponents can help you to simplify the expression by combining the exponents and canceling out common factors.

Q: What are some additional resources for learning about simplifying expressions?

A: Some additional resources for learning about simplifying expressions include:

• Algebraic Manipulation, MIT OpenCourseWare
• Exponents, Wolfram MathWorld
• Simplifying Expressions, Purplemath
• Algebraic Manipulation, Khan Academy

Conclusion

In this article, we have answered some of the most frequently asked questions about simplifying expressions and provided additional resources for further learning. We hope that this article has been helpful in providing you with a better understanding of how to simplify complex expressions and apply this skill to various mathematical problems.

Final Answer

The final answer is $\boxed{\frac{5}{9}}$.

Frequently Asked Questions

• What is the process of simplifying an expression?
• What are the properties of exponents that we used to simplify the expression?
• What is the final answer to the expression?
• How do I simplify a complex expression?
• What are some common mistakes to avoid when simplifying expressions?
• How do I know when to use the properties of exponents?
• What are some additional resources for learning about simplifying expressions?

References

• [1] Algebraic Manipulation, Wikipedia
• [2] Exponents, Khan Academy
• [3] Simplifying Expressions, Mathway

Further Reading

• Algebraic Manipulation, MIT OpenCourseWare
• Exponents, Wolfram MathWorld
• Simplifying Expressions, Purplemath
• Algebraic Manipulation, Khan Academy