Simplify The Expression:${ \frac{10^{2x+4} \cdot 4 {1-x}}{25 {2+x}} }$

Mar 9, 2025 by ADMIN 72 views

Simplify the Expression: A Comprehensive Guide to Algebraic Manipulation

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 the expression $\frac{10^{2x+4} \cdot 4^{1-x}}{25^{2+x}}$ . We will break down the expression into manageable parts, apply various algebraic manipulations, and ultimately arrive at a simplified form.

Understanding the Expression

Before we dive into simplifying the expression, let's take a closer look at its components. The expression consists of three main parts:

$10^{2x+4}$ : This is an exponential term with a base of 10 and an exponent of $2x+4$ .
$4^{1-x}$ : This is another exponential term with a base of 4 and an exponent of $1-x$ .
$25^{2+x}$ : This is the denominator of the expression, with a base of 25 and an exponent of $2+x$ .

Simplifying the Expression

To simplify the expression, we can start by applying the properties of exponents. Specifically, we can use the property that states $a^{m+n} = a^m \cdot a^n$ .

Step 1: Simplify the Numerator

Let's start by simplifying the numerator of the expression. We can rewrite $10^{2x+4}$ as $10^{2x} \cdot 10^4$ . Similarly, we can rewrite $4^{1-x}$ as $4^{1} \cdot 4^{-x}$ .

import sympy as sp
x = sp.symbols('x')

numerator = (10**(2*x + 4)) * (4**(1 - x))
simplified_numerator = sp.simplify(numerator)
print(simplified_numerator)

Step 2: Simplify the Denominator

Next, let's simplify the denominator of the expression. We can rewrite $25^{2+x}$ as $25^{2} \cdot 25^x$ .

# Simplify the denominator
denominator = (25**(2 + x))
simplified_denominator = sp.simplify(denominator)
print(simplified_denominator)

Step 3: Combine the Simplified Numerator and Denominator

Now that we have simplified the numerator and denominator, we can combine them to get the simplified expression.

# Combine the simplified numerator and denominator
simplified_expression = simplified_numerator / simplified_denominator
print(simplified_expression)

Final Simplified Expression

After applying the properties of exponents and simplifying the numerator and denominator, we arrive at the final simplified expression:

\frac{10^{2x} \cdot 10^4 \cdot 4^{1} \cdot 4^{-x}}{25^{2} \cdot 25^x}

Further Simplification

We can further simplify the expression by canceling out common factors in the numerator and denominator.

# Cancel out common factors
final_simplified_expression = sp.cancel(simplified_expression)
print(final_simplified_expression)

Conclusion

In this article, we simplified the expression $\frac{10^{2x+4} \cdot 4^{1-x}}{25^{2+x}}$ using various algebraic manipulations. We applied the properties of exponents, simplified the numerator and denominator, and arrived at the final simplified expression. This expression can be further simplified by canceling out common factors.

Final Answer

The final simplified expression is:

\frac{10^{2x} \cdot 4^{1-x}}{5^{2+2x}}

This expression is the final answer to the problem.

Additional Tips and Tricks

When simplifying expressions, it's essential to apply the properties of exponents correctly.
Simplifying the numerator and denominator separately can make the process more manageable.
Canceling out common factors can help simplify the expression further.

By following these tips and tricks, you can simplify complex expressions and arrive at the final answer with ease.

Frequently Asked Questions

Q: What is the final simplified expression?
A: The final simplified expression is $\frac{10^{2x} \cdot 4^{1-x}}{5^{2+2x}}$ .
Q: How do I simplify complex expressions?
A: To simplify complex expressions, apply the properties of exponents, simplify the numerator and denominator separately, and cancel out common factors.

By following these steps and tips, you can simplify complex expressions and arrive at the final answer with ease.

Introduction

In our previous article, we simplified the expression $\frac{10^{2x+4} \cdot 4^{1-x}}{25^{2+x}}$ using various algebraic manipulations. We applied the properties of exponents, simplified the numerator and denominator, and arrived at the final simplified expression. In this article, we will answer some frequently asked questions related to simplifying expressions.

Q&A

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{10^{2x} \cdot 4^{1-x}}{5^{2+2x}}$ .

Q: How do I simplify complex expressions?

A: To simplify complex expressions, apply the properties of exponents, simplify the numerator and denominator separately, and cancel out common factors.

Q: What are some common properties of exponents that I should know?

A: Some common properties of exponents that you should know include:

$a^{m+n} = a^m \cdot a^n$
$a^{m-n} = \frac{a^m}{a^n}$
$(a^m)^n = a^{m \cdot n}$

Q: How do I apply the properties of exponents to simplify expressions?

A: To apply the properties of exponents to simplify expressions, follow these steps:

Identify the properties of exponents that can be applied to the expression.
Simplify the numerator and denominator separately using the properties of exponents.
Cancel out common factors in the numerator and denominator.

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

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

Not applying the properties of exponents correctly.
Not simplifying the numerator and denominator separately.
Not canceling out common factors in the numerator and denominator.

Q: How do I check my work when simplifying expressions?

A: To check your work when simplifying expressions, follow these steps:

Plug in a value for the variable (e.g. x = 0).
Simplify the expression using the value you plugged in.
Check if the simplified expression is correct.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, including:

Calculating interest rates on loans.
Determining the cost of goods sold.
Calculating the area and perimeter of shapes.

Conclusion

Simplifying expressions is an essential skill in algebra and mathematics. By applying the properties of exponents, simplifying the numerator and denominator separately, and canceling out common factors, you can simplify complex expressions and arrive at the final answer with ease. Remember to check your work and avoid common mistakes when simplifying expressions.

Additional Tips and Tricks

Practice simplifying expressions regularly to build your skills and confidence.
Use online resources and tools to help you simplify expressions.
Break down complex expressions into smaller, more manageable parts.

By following these tips and tricks, you can become a master of simplifying expressions and tackle even the most complex problems with ease.

Frequently Asked Questions

Q: What are some common properties of exponents that I should know?
A: Some common properties of exponents that you should know include $a^{m+n} = a^m \cdot a^n$ , $a^{m-n} = \frac{a^m}{a^n}$ , and $(a^m)^n = a^{m \cdot n}$ .
Q: How do I apply the properties of exponents to simplify expressions?
A: To apply the properties of exponents to simplify expressions, follow these steps: identify the properties of exponents that can be applied to the expression, simplify the numerator and denominator separately using the properties of exponents, and cancel out common factors in the numerator and denominator.

By following these steps and tips, you can simplify complex expressions and arrive at the final answer with ease.