Simplify The Expression:$\[ \frac{3^{3x-1} \cdot 5^{x-3}}{45^{x-2}} \\]

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. In this article, we will focus on simplifying the given expression, which involves exponents and fractions. We will use various techniques, such as factoring and canceling, to simplify the expression and arrive at a more manageable form.

Understanding the Expression

The given expression is $\frac{3^{3x-1} \cdot 5^{x-3}}{45^{x-2}}$. This expression involves exponents, fractions, and variables. To simplify this expression, we need to understand the properties of exponents and fractions.

Properties of Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, $a^b$ means $a$ multiplied by itself $b$ times. The properties of exponents are as follows:

• Product of Powers: When multiplying two powers with the same base, we add the exponents. For example, $a^b \cdot a^c = a^{b+c}$.
• Power of a Power: When raising a power to another power, we multiply the exponents. For example, $(a^b)^c = a^{bc}$.
• Zero Exponent: Any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.

Properties of Fractions

Fractions are a way of representing part of a whole. The properties of fractions are as follows:

• Multiplication of Fractions: When multiplying two fractions, we multiply the numerators and denominators separately. For example, $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$.
• Division of Fractions: When dividing two fractions, we invert the second fraction and multiply. For example, $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc}$.

Simplifying the Expression

Now that we have a good understanding of the properties of exponents and fractions, we can simplify the given expression.

Step 1: Factor the Denominator

The denominator of the expression is $45^{x-2}$. We can factor this expression as $45^{x-2} = (3 \cdot 5)^{x-2} = 3^{x-2} \cdot 5^{x-2}$.

Step 2: Rewrite the Expression

Using the factored form of the denominator, we can rewrite the expression as $\frac{3^{3x-1} \cdot 5^{x-3}}{3^{x-2} \cdot 5^{x-2}}$.

Step 3: Cancel Common Factors

Now that we have rewritten the expression, we can cancel common factors. The common factors are $3^{x-2}$ and $5^{x-2}$. Canceling these factors, we get $\frac{3^{3x-1}}{3^{x-2}} \cdot \frac{5^{x-3}}{5^{x-2}}$.

Step 4: Simplify the Expression

Using the properties of exponents, we can simplify the expression further. The expression becomes $3^{3x-1-(x-2)} \cdot 5^{x-3-(x-2)}$.

Step 5: Evaluate the Exponents

Evaluating the exponents, we get $3^{2x+1} \cdot 5^{1}$.

Step 6: Simplify the Expression

The expression $3^{2x+1} \cdot 5^{1}$ can be simplified further by multiplying the terms. The expression becomes $3^{2x+1} \cdot 5$.

Conclusion

In this article, we simplified the given expression using various techniques, such as factoring and canceling. We used the properties of exponents and fractions to arrive at a more manageable form. The final simplified expression is $3^{2x+1} \cdot 5$.

Final Answer

The final answer is $\boxed{3^{2x+1} \cdot 5}$.

Discussion

The given expression can be simplified using various techniques. The key to simplifying the expression is to understand the properties of exponents and fractions. By factoring the denominator and canceling common factors, we can arrive at a more manageable form. The final simplified expression is $3^{2x+1} \cdot 5$.

Related Topics

• Simplifying Algebraic Expressions: Simplifying algebraic expressions is a crucial skill in mathematics. It requires a deep understanding of the underlying concepts and the ability to apply various techniques, such as factoring and canceling.
• Properties of Exponents: Exponents are a shorthand way of writing repeated multiplication. The properties of exponents are as follows: product of powers, power of a power, and zero exponent.
• Properties of Fractions: Fractions are a way of representing part of a whole. The properties of fractions are as follows: multiplication of fractions and division of fractions.

References

• Algebraic Expressions: Algebraic expressions are a way of representing mathematical relationships using variables and constants. They can be simplified using various techniques, such as factoring and canceling.
• Exponents: Exponents are a shorthand way of writing repeated multiplication. They have various properties, such as product of powers, power of a power, and zero exponent.
• Fractions: Fractions are a way of representing part of a whole. They have various properties, such as multiplication of fractions and division of fractions.
  

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Introduction

In our previous article, we simplified the given expression $\frac{3^{3x-1} \cdot 5^{x-3}}{45^{x-2}}$ using various techniques, such as factoring and canceling. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

Q1: What is the final simplified expression?

A1: The final simplified expression is $3^{2x+1} \cdot 5$.

Q2: How do I simplify the expression $\frac{3^{3x-1} \cdot 5^{x-3}}{45^{x-2}}$?

A2: To simplify the expression, you can follow these steps:

• Factor the denominator as $45^{x-2} = (3 \cdot 5)^{x-2} = 3^{x-2} \cdot 5^{x-2}$.
• Rewrite the expression as $\frac{3^{3x-1} \cdot 5^{x-3}}{3^{x-2} \cdot 5^{x-2}}$.
• Cancel common factors, which are $3^{x-2}$ and $5^{x-2}$.
• Simplify the expression further using the properties of exponents.
• Evaluate the exponents and simplify the expression.

Q3: What are the properties of exponents?

A3: The properties of exponents are as follows:

• Product of Powers: When multiplying two powers with the same base, we add the exponents. For example, $a^b \cdot a^c = a^{b+c}$.
• Power of a Power: When raising a power to another power, we multiply the exponents. For example, $(a^b)^c = a^{bc}$.
• Zero Exponent: Any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.

Q4: What are the properties of fractions?

A4: The properties of fractions are as follows:

• Multiplication of Fractions: When multiplying two fractions, we multiply the numerators and denominators separately. For example, $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$.
• Division of Fractions: When dividing two fractions, we invert the second fraction and multiply. For example, $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc}$.

Q5: How do I factor the denominator $45^{x-2}$?

A5: To factor the denominator $45^{x-2}$, you can express it as $(3 \cdot 5)^{x-2} = 3^{x-2} \cdot 5^{x-2}$.

Q6: How do I cancel common factors in the expression $\frac{3^{3x-1} \cdot 5^{x-3}}{3^{x-2} \cdot 5^{x-2}}$?

A6: To cancel common factors, you can divide the numerator and denominator by the common factors, which are $3^{x-2}$ and $5^{x-2}$.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression $\frac{3^{3x-1} \cdot 5^{x-3}}{45^{x-2}}$. We provided step-by-step instructions on how to simplify the expression and answered questions related to the properties of exponents and fractions.

Final Answer

The final answer is $\boxed{3^{2x+1} \cdot 5}$.

Discussion

The given expression can be simplified using various techniques, such as factoring and canceling. The key to simplifying the expression is to understand the properties of exponents and fractions. By factoring the denominator and canceling common factors, we can arrive at a more manageable form. The final simplified expression is $3^{2x+1} \cdot 5$.

Related Topics

• Simplifying Algebraic Expressions: Simplifying algebraic expressions is a crucial skill in mathematics. It requires a deep understanding of the underlying concepts and the ability to apply various techniques, such as factoring and canceling.
• Properties of Exponents: Exponents are a shorthand way of writing repeated multiplication. The properties of exponents are as follows: product of powers, power of a power, and zero exponent.
• Properties of Fractions: Fractions are a way of representing part of a whole. The properties of fractions are as follows: multiplication of fractions and division of fractions.

References

• Algebraic Expressions: Algebraic expressions are a way of representing mathematical relationships using variables and constants. They can be simplified using various techniques, such as factoring and canceling.
• Exponents: Exponents are a shorthand way of writing repeated multiplication. They have various properties, such as product of powers, power of a power, and zero exponent.
• Fractions: Fractions are a way of representing part of a whole. They have various properties, such as multiplication of fractions and division of fractions.