Simplify The Expression:$\[ \frac{2 \cdot 3^x}{3^{x+2} - 3^x} \\]

Feb 25, 2025 by ADMIN 66 views

**Simplify the Expression: A Step-by-Step Guide** =====================================================

Introduction

In this article, we will simplify the given expression $\frac{2 \cdot 3^x}{3^{x+2} - 3^x}$ . This expression involves exponents and fractions, and simplifying it requires a clear understanding of exponent rules and fraction simplification techniques.

Understanding the Expression

The given expression is $\frac{2 \cdot 3^x}{3^{x+2} - 3^x}$ . To simplify this expression, we need to understand the properties of exponents and fractions.

Exponent Rules

Exponents are a shorthand way of writing repeated multiplication. For example, $3^x$ means $3$ multiplied by itself $x$ times. When we have an expression like $3^{x+2}$ , we can use the exponent rule to simplify it.

Fraction Simplification

Fractions are a way of representing a part of a whole. To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD.

Step 1: Simplify the Exponents

The first step in simplifying the expression is to simplify the exponents. We can use the exponent rule to rewrite $3^{x+2}$ as $3^x \cdot 3^2$ .

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the expression
expr = (2 * 3**x) / (3**(x+2) - 3**x)

# Simplify the exponents
simplified_expr = expr.subs(3**(x+2), 3**x * 3**2)

print(simplified_expr)

Step 2: Factor the Denominator

The next step is to factor the denominator. We can use the difference of squares formula to factor $3^x \cdot 3^2 - 3^x$ .

# Factor the denominator
factored_denominator = sp.factor(3**x * 3**2 - 3**x)

print(factored_denominator)

Step 3: Cancel Common Factors

Now that we have factored the denominator, we can cancel common factors between the numerator and denominator.

# Cancel common factors
canceled_expr = simplified_expr.cancel()

print(canceled_expr)

Step 4: Simplify the Expression

The final step is to simplify the expression. We can use the simplified expression from Step 3 to simplify the expression.

# Simplify the expression
simplified_expr = canceled_expr.simplify()

print(simplified_expr)

Conclusion

In this article, we simplified the expression $\frac{2 \cdot 3^x}{3^{x+2} - 3^x}$ using exponent rules and fraction simplification techniques. We used Python code to simplify the expression step-by-step.

Q&A

Q: What is the simplified expression?

A: The simplified expression is $\frac{2}{3^2 - 1}$ .

Q: How do I simplify the expression using exponent rules?

A: To simplify the expression using exponent rules, you need to rewrite $3^{x+2}$ as $3^x \cdot 3^2$ .

Q: How do I factor the denominator?

A: To factor the denominator, you can use the difference of squares formula to factor $3^x \cdot 3^2 - 3^x$ .

Q: How do I cancel common factors?

A: To cancel common factors, you need to find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD.

Q: How do I simplify the expression?

A: To simplify the expression, you need to use the simplified expression from Step 3 and simplify it further.

Common Mistakes

Mistake 1: Not simplifying the exponents

Not simplifying the exponents can lead to a more complex expression.

Mistake 2: Not factoring the denominator

Not factoring the denominator can lead to a more complex expression.

Mistake 3: Not canceling common factors

Not canceling common factors can lead to a more complex expression.

Mistake 4: Not simplifying the expression

Not simplifying the expression can lead to a more complex expression.

Best Practices

Best Practice 1: Simplify the exponents

Simplifying the exponents can lead to a simpler expression.

Best Practice 2: Factor the denominator

Factoring the denominator can lead to a simpler expression.

Best Practice 3: Cancel common factors

Canceling common factors can lead to a simpler expression.

Best Practice 4: Simplify the expression

Simplifying the expression can lead to a simpler expression.