Simplify The Expression: $\[ \frac{45x^5y^3}{18xy^3} \\]

Feb 27, 2025 by ADMIN 57 views

**Simplify the Expression: A Step-by-Step Guide to Algebraic Manipulation**

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements. Simplifying an expression involves reducing it to its simplest form, which can be done by combining like terms, canceling out common factors, and rearranging the terms. In this article, we will simplify the expression $\frac{45x^5y^3}{18xy^3}$ using a step-by-step approach.

Understanding the Expression

The given expression is a fraction with two terms in the numerator and one term in the denominator. The numerator is $45x^5y^3$ , and the denominator is $18xy^3$ . To simplify this expression, we need to understand the properties of exponents and fractions.

Step 1: Factorize the Numerator and Denominator

The first step in simplifying the expression is to factorize the numerator and denominator. We can factorize $45$ as $9 \times 5$ , $x^5$ as $x^4 \times x$ , and $y^3$ as $y^2 \times y$ . Similarly, we can factorize $18$ as $9 \times 2$ , $x$ as $x^1$ , and $y^3$ as $y^2 \times y$ .

\frac{45x^5y^3}{18xy^3} = \frac{9 \times 5 \times x^4 \times x \times y^2 \times y}{9 \times 2 \times x \times y^2 \times y}

Step 2: Cancel Out Common Factors

Now that we have factorized the numerator and denominator, we can cancel out common factors. The common factors between the numerator and denominator are $9$ , $x$ , and $y^2$ . We can cancel out these factors by dividing them out.

\frac{9 \times 5 \times x^4 \times x \times y^2 \times y}{9 \times 2 \times x \times y^2 \times y} = \frac{5 \times x^4}{2}

Step 3: Simplify the Expression

Now that we have canceled out the common factors, we can simplify the expression further. We can simplify the expression by combining the remaining terms.

\frac{5 \times x^4}{2} = \frac{5x^4}{2}

Conclusion

In this article, we simplified the expression $\frac{45x^5y^3}{18xy^3}$ using a step-by-step approach. We factorized the numerator and denominator, canceled out common factors, and simplified the expression further. The simplified expression is $\frac{5x^4}{2}$ . This expression is in its simplest form, and it cannot be simplified further.

Tips and Tricks

When simplifying expressions, always look for common factors between the numerator and denominator.
Use the properties of exponents to simplify expressions.
Cancel out common factors by dividing them out.
Simplify the expression further by combining the remaining terms.

Common Mistakes to Avoid

Not factorizing the numerator and denominator properly.
Not canceling out common factors correctly.
Not simplifying the expression further after canceling out common factors.

Real-World Applications

Simplifying expressions is a crucial skill in algebra that has many real-world applications. Some of the real-world applications of simplifying expressions include:

Solving equations and manipulating mathematical statements.
Calculating rates and ratios.
Determining the area and perimeter of shapes.
Finding the volume and surface area of solids.

Conclusion

Introduction

In our previous article, we simplified the expression $\frac{45x^5y^3}{18xy^3}$ using a step-by-step approach. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions.

Q&A

Q: What is the purpose of simplifying expressions?

A: The purpose of simplifying expressions is to reduce them to their simplest form, which can be done by combining like terms, canceling out common factors, and rearranging the terms. Simplifying expressions helps us solve equations and manipulate mathematical statements.

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

A: To simplify an expression with multiple variables, you need to understand the properties of exponents and fractions. You can factorize the numerator and denominator, cancel out common factors, and simplify the expression further by combining the remaining terms.

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

A: Some common mistakes to avoid when simplifying expressions include not factorizing the numerator and denominator properly, not canceling out common factors correctly, and not simplifying the expression further after canceling out common factors.

Q: How do I determine if an expression is in its simplest form?

A: To determine if an expression is in its simplest form, you need to check if it cannot be simplified further by combining like terms, canceling out common factors, or rearranging the terms.

Q: Can I simplify an expression with negative exponents?

A: Yes, you can simplify an expression with negative exponents. To simplify an expression with negative exponents, you need to understand the properties of exponents and fractions. You can rewrite the negative exponent as a positive exponent by taking the reciprocal of the base.

Q: How do I simplify an expression with fractions in the numerator and denominator?

A: To simplify an expression with fractions in the numerator and denominator, you need to understand the properties of fractions and exponents. You can simplify the fractions by canceling out common factors and then simplify the expression further by combining the remaining terms.

Q: Can I simplify an expression with variables in the denominator?

A: Yes, you can simplify an expression with variables in the denominator. To simplify an expression with variables in the denominator, you need to understand the properties of fractions and exponents. You can simplify the expression by canceling out common factors and then simplify the expression further by combining the remaining terms.

Q: How do I simplify an expression with multiple fractions in the numerator and denominator?

A: To simplify an expression with multiple fractions in the numerator and denominator, you need to understand the properties of fractions and exponents. You can simplify the fractions by canceling out common factors and then simplify the expression further by combining the remaining terms.

Q: Can I simplify an expression with absolute values?

A: Yes, you can simplify an expression with absolute values. To simplify an expression with absolute values, you need to understand the properties of absolute values and exponents. You can simplify the expression by canceling out common factors and then simplify the expression further by combining the remaining terms.

Conclusion

In conclusion, simplifying expressions is a crucial skill in algebra that helps us solve equations and manipulate mathematical statements. By following a step-by-step approach and understanding the properties of exponents and fractions, we can simplify expressions and arrive at their simplest form. The FAQs in this article provide additional guidance on simplifying expressions and help to clarify common mistakes to avoid.

Tips and Tricks

Always look for common factors between the numerator and denominator.
Use the properties of exponents to simplify expressions.
Cancel out common factors by dividing them out.
Simplify the expression further by combining the remaining terms.
Understand the properties of fractions and exponents to simplify expressions with multiple variables, negative exponents, fractions in the numerator and denominator, variables in the denominator, multiple fractions in the numerator and denominator, and absolute values.

Real-World Applications

Simplifying expressions is a crucial skill in algebra that has many real-world applications. Some of the real-world applications of simplifying expressions include:

Solving equations and manipulating mathematical statements.
Calculating rates and ratios.
Determining the area and perimeter of shapes.
Finding the volume and surface area of solids.
Simplifying expressions with multiple variables, negative exponents, fractions in the numerator and denominator, variables in the denominator, multiple fractions in the numerator and denominator, and absolute values.