\[$(2 \sqrt{12} + \sqrt{24})^2\$\] Can Be Expressed In Simplest Form As \[$a + B \sqrt{c}\$\].The Value Of \[$abc\$\] Is \[$\qquad\$\] -
Introduction
Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression {(2 \sqrt{12} + \sqrt{24})^2$}$. We will break down the expression into its simplest form and calculate the value of {abc$}$.
Understanding Radical Expressions
A radical expression is a mathematical expression that contains a square root or a higher root of a number. Radical expressions can be simplified by factoring out perfect squares from the radicand (the number inside the radical sign). This process involves identifying the largest perfect square that divides the radicand and then simplifying the expression.
Simplifying the Given Expression
To simplify the given expression {(2 \sqrt{12} + \sqrt{24})^2$}$, we need to start by simplifying the individual terms inside the parentheses.
Simplifying ${$2 \sqrt{12}$]
We can simplify [$2 \sqrt{12}$] by factoring out the perfect square from the radicand.
[$2 \sqrt{12} = 2 \sqrt{4 \cdot 3} = 2 \cdot 2 \sqrt{3} = 4 \sqrt{3}$]
Simplifying [$\sqrt{24}$]
We can simplify [$\sqrt{24}$] by factoring out the perfect square from the radicand.
[$\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2 \sqrt{6}$]
Simplifying the Expression Inside the Parentheses
Now that we have simplified the individual terms, we can simplify the expression inside the parentheses.
[$2 \sqrt{12} + \sqrt{24} = 4 \sqrt{3} + 2 \sqrt{6}$]
Squaring the Expression
To simplify the given expression [(2 \sqrt{12} + \sqrt{24})^2\$}, we need to square the expression inside the parentheses.
{(2 \sqrt{12} + \sqrt{24})^2 = (4 \sqrt{3} + 2 \sqrt{6})^2$]
Using the formula [(a + b)^2 = a^2 + 2ab + b^2\$}, we can expand the squared expression.
{(4 \sqrt{3} + 2 \sqrt{6})^2 = (4 \sqrt{3})^2 + 2(4 \sqrt{3})(2 \sqrt{6}) + (2 \sqrt{6})^2$]
Simplifying the expression further, we get:
[$(4 \sqrt{3})^2 = 16 \cdot 3 = 48$]
[$2(4 \sqrt{3})(2 \sqrt{6}) = 2 \cdot 4 \cdot 2 \sqrt{3} \cdot \sqrt{6} = 16 \sqrt{18} = 16 \sqrt{9 \cdot 2} = 16 \cdot 3 \sqrt{2} = 48 \sqrt{2}$]
[$(2 \sqrt{6})^2 = 4 \cdot 6 = 24$]
Combining the simplified terms, we get:
[$(4 \sqrt{3} + 2 \sqrt{6})^2 = 48 + 48 \sqrt{2} + 24 = 72 + 48 \sqrt{2}$]
Simplifying the Expression
The simplified expression is [a + b \sqrt{c}$], where [b = 48$, and [abc$}$ is ${$72 \cdot 48 \cdot 2 = 6912$].
Conclusion
In this article, we simplified the radical expression [(2 \sqrt{12} + \sqrt{24})^2\$} and calculated the value of {abc$}$. We started by simplifying the individual terms inside the parentheses and then squared the expression. Finally, we simplified the resulting expression and calculated the value of {abc$}$. This process demonstrates the importance of simplifying radical expressions and understanding the properties of square roots.
Final Answer
Introduction
Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In our previous article, we explored the process of simplifying radical expressions, with a focus on the given expression {(2 \sqrt{12} + \sqrt{24})^2$}$. In this article, we will answer some frequently asked questions about simplifying radical expressions.
Q&A
Q: What is the difference between a radical expression and a rational expression?
A: A radical expression is a mathematical expression that contains a square root or a higher root of a number. A rational expression, on the other hand, is a mathematical expression that contains a fraction with a polynomial in the numerator and a polynomial in the denominator.
Q: How do I simplify a radical expression?
A: To simplify a radical expression, you need to factor out perfect squares from the radicand (the number inside the radical sign). This process involves identifying the largest perfect square that divides the radicand and then simplifying the expression.
Q: What is the formula for simplifying a radical expression?
A: The formula for simplifying a radical expression is:
[$\sqrt{a} = \sqrt{b \cdot c} = \sqrt{b} \cdot \sqrt{c}$]
where [b$, and [$c$] are positive integers.
Q: How do I simplify an expression with multiple radical terms?
A: To simplify an expression with multiple radical terms, you need to combine the terms using the distributive property. This involves multiplying each term by the other terms and then simplifying the resulting expression.
Q: What is the difference between a rationalized denominator and a simplified radical expression?
A: A rationalized denominator is a radical expression with a rational denominator, while a simplified radical expression is a radical expression with a simplified radicand.
Q: How do I rationalize a denominator?
A: To rationalize a denominator, you need to multiply the numerator and denominator by the conjugate of the denominator. This involves multiplying the numerator and denominator by the same expression, but with the opposite sign.
Q: What is the formula for rationalizing a denominator?
A: The formula for rationalizing a denominator is:
[$\frac{\sqrt{a}}{\sqrt{b}} = \frac{\sqrt{a} \cdot \sqrt{b}}{\sqrt{b} \cdot \sqrt{b}} = \frac{\sqrt{ab}}{b}$]
where [b$, and [$c$] are positive integers.
Q: How do I simplify an expression with a radical in the denominator?
A: To simplify an expression with a radical in the denominator, you need to rationalize the denominator. This involves multiplying the numerator and denominator by the conjugate of the denominator.
Q: What is the difference between a simplified radical expression and a simplified rational expression?
A: A simplified radical expression is a radical expression with a simplified radicand, while a simplified rational expression is a rational expression with a simplified numerator and denominator.
Q: How do I simplify a rational expression?
A: To simplify a rational expression, you need to factor the numerator and denominator and then cancel out any common factors.
Q: What is the formula for simplifying a rational expression?
A: The formula for simplifying a rational expression is:
[$\frac{a}{b} = \frac{c}{d}$]
where [b$, [d$] are positive integers.
Conclusion
In this article, we answered some frequently asked questions about simplifying radical expressions. We covered topics such as the difference between a radical expression and a rational expression, how to simplify a radical expression, and how to rationalize a denominator. We also provided formulas and examples to help illustrate the concepts.
Final Answer
The final answer is: