11. Simplify The Expression:$\[ \frac{\left(24 X^7 Y^{11}\right)}{\left(2 X^4 Y^5\right)^2} \\]

Feb 26, 2025 by ADMIN 96 views

$11. Simplify the expression: $\frac{\left(24 x^7 y^{11}\right)}{\left(2 x^4 y^5\right)^2}$$

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved in simplifying expressions. In this article, we will focus on simplifying the given expression using the rules of exponents and algebraic manipulation.

Understanding the Expression

The given expression is $\frac{\left(24 x^7 y^{11}\right)}{\left(2 x^4 y^5\right)^2}$ . To simplify this expression, we need to understand the rules of exponents and how to manipulate them. The expression involves the division of two algebraic expressions, and we need to simplify it by applying the rules of exponents.

Simplifying the Expression

To simplify the expression, we can start by applying the rule of exponents that states $\left(a^m\right)^n = a^{m \cdot n}$ . Using this rule, we can rewrite the expression as $\frac{24 x^7 y^{11}}{2^2 \cdot x^{4 \cdot 2} \cdot y^{5 \cdot 2}}$ .

Applying the Rule of Exponents

Now, we can apply the rule of exponents to simplify the expression further. We can rewrite the expression as $\frac{24 x^7 y^{11}}{4 \cdot x^8 \cdot y^{10}}$ .

Canceling Out Common Factors

We can simplify the expression further by canceling out common factors. We can cancel out the common factor of 4 in the numerator and denominator, as well as the common factor of $x^7$ in the numerator and denominator.

Simplifying the Expression

After canceling out the common factors, we are left with $\frac{6 y}{x}$ . This is the simplified form of the given expression.

Conclusion

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

Understand the rules of exponents: The rules of exponents are crucial in simplifying algebraic expressions. Make sure you understand the rules of exponents, including the rule that states $\left(a^m\right)^n = a^{m \cdot n}$ .
Apply the rule of exponents: Once you understand the rules of exponents, you can apply them to simplify the expression.
Cancel out common factors: Canceling out common factors is an essential step in simplifying algebraic expressions. Make sure you cancel out common factors in the numerator and denominator.
Simplify the expression: After canceling out common factors, you can simplify the expression further by applying the rules of exponents and algebraic manipulation.

Examples

Here are some examples of simplifying algebraic expressions:

Example 1: Simplify the expression $\frac{\left(36 x^5 y^3\right)}{\left(6 x^2 y\right)^2}$ .
Example 2: Simplify the expression $\frac{\left(24 x^7 y^{11}\right)}{\left(2 x^4 y^5\right)^2}$ .
Example 3: Simplify the expression $\frac{\left(48 x^3 y^2\right)}{\left(8 x y\right)^2}$ .

Solutions

Here are the solutions to the examples:

Example 1: The simplified form of the expression is $\frac{6 x^3 y^2}{x^4 y^2}$ .
Example 2: The simplified form of the expression is $\frac{6 y}{x}$ .
Example 3: The simplified form of the expression is $\frac{6 x^2 y}{x^2 y}$ .

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved in simplifying expressions. By applying the rules of exponents and algebraic manipulation, we can simplify complex expressions and arrive at a simplified form. Remember to understand the rules of exponents, apply the rule of exponents, cancel out common factors, and simplify the expression to arrive at a simplified form.

Introduction

In the previous article, we simplified the expression $\frac{\left(24 x^7 y^{11}\right)}{\left(2 x^4 y^5\right)^2}$ using the rules of exponents and algebraic manipulation. In this article, we will answer some frequently asked questions (FAQs) related to simplifying algebraic expressions.

Q&A

Q1: What are the rules of exponents?

A1: The rules of exponents are a set of rules that govern the behavior of exponents in algebraic expressions. The main rules of exponents are:

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

Q2: How do I simplify an algebraic expression?

A2: To simplify an algebraic expression, you need to apply the rules of exponents and algebraic manipulation. Here are the steps to simplify an algebraic expression:

Understand the rules of exponents: Make sure you understand the rules of exponents, including the rule that states $\left(a^m\right)^n = a^{m \cdot n}$ .
Apply the rule of exponents: Once you understand the rules of exponents, you can apply them to simplify the expression.
Cancel out common factors: Cancel out common factors in the numerator and denominator.
Simplify the expression: After canceling out common factors, you can simplify the expression further by applying the rules of exponents and algebraic manipulation.

Q3: What is the difference between a variable and a constant?

A3: A variable is a letter or symbol that represents a value that can change. A constant is a value that does not change. For example, in the expression $x^2 + 3$ , $x$ is a variable and $3$ is a constant.

Q4: How do I simplify an expression with negative exponents?

A4: To simplify an expression with negative exponents, you need to apply the rule that states $\frac{1}{a^m} = a^{-m}$ . For example, to simplify the expression $\frac{1}{x^2}$ , you can rewrite it as $x^{-2}$ .

Q5: What is the difference between a rational expression and an irrational expression?

A5: A rational expression is an expression that can be written in the form $\frac{a}{b}$ , where $a$ and $b$ are integers. An irrational expression is an expression that cannot be written in the form $\frac{a}{b}$ , where $a$ and $b$ are integers. For example, the expression $\frac{1}{x}$ is a rational expression, while the expression $\sqrt{x}$ is an irrational expression.