Problem 83 - Algebra 2

Question

Rationalize the denominator of each expression. Assume that all variables are positive. \(\frac{\sqrt{36 x^{3}}}{\sqrt{12 x}}\)

Step-by-Step Solution

Answer

The rationalized form of the given expression is \( 3\sqrt{3x} \).

Step 1: Simplify Both Square Roots

First, simplify both square roots in the expression \(\frac{\sqrt{36 x^{3}}}{\sqrt{12 x}}\). The square roots simplify to \( \frac{6x\sqrt{x}}{2\sqrt{3x}} \).

Step 2: Simplify the Fractions Along X variable

Next, simplify the fraction in terms of x. This reduces the expression to \( \frac{3\sqrt{x}}{\sqrt{3}} \).

Step 3: Rationalize the Denominator

Lastly, in order to rationalize the denominator, multiply both the numerator and denominator by \(\sqrt{3}\). This results in the expression \( 3\sqrt{3x} \).

Key Concepts

simplifying square roots

When we talk about simplifying square roots, we refer to the process of expressing them in their simplest form. With the expression \( \sqrt{36x^3} \), our first task is to look inside the square root to identify perfect squares.

• Identify Perfect Squares: Perfect squares are numbers or variables that have a whole number as their square root. In this case, \(36\) is a perfect square because \(6^2 = 36\), and \(x^2\) is also a perfect square.
• Simplify Inside the Root: \(\sqrt{36} = 6\) and \(\sqrt{x^2} = x\), so the simplified form of \(\sqrt{36x^3}\) becomes \(6x\sqrt{x}\).

The same process is applied to the denominator \(\sqrt{12x}\), where we identify factors of \(12\) such as \(4 \, (2^2)\), thus simplifying \(\sqrt{12} = 2\sqrt{3}\). Therefore, \(\sqrt{12x} = 2\sqrt{3x}\). Breaking down square roots helps in making further calculations easier and clearer.

algebraic expressions

An algebraic expression consists of numbers, variables, and arithmetic operations. They can be as simple as just a variable, like \(x\), or a more complex combination like \( \frac{6x\sqrt{x}}{2\sqrt{3x}} \).

• Terms: An algebraic expression may have multiple terms, separated by addition or subtraction. Each term can consist of products or divisions of variables and numbers.
• Simplification: The goal when working with algebraic expressions is to simplify them as much as possible. In our example, \( \frac{6x\sqrt{x}}{2\sqrt{3x}} \), by dividing the coefficients and canceling common factors in terms of \(x\), we simplify it to \( \frac{3\sqrt{x}}{\sqrt{3}} \).

Simplification makes these expressions easier to interpret and use in further calculations or evaluations.

rational expressions

Rational expressions are fractions in which both the numerator and the denominator are polynomials. Our example, \( \frac{3\sqrt{x}}{\sqrt{3}} \), can be considered a rational expression with a radical in the denominator.

• Rationalizing the Denominator: To simplify a rational expression, it's often necessary to remove any radicals from the denominator. This process is called rationalizing the denominator.
• Multiplication: Multiply both the numerator and the denominator by the necessary square root to eliminate the radical in the denominator. With \( \frac{3\sqrt{x}}{\sqrt{3}} \), multiply both by \(\sqrt{3}\) to achieve \(\frac{3\sqrt{3x}}{3}\).

This reduces to \(\sqrt{3x}\), a rational expression without any radicals in the denominator. Rationalizing not only simplifies the expression but also makes it more suitable for further mathematical operations.