Problem 11 - Elementary Algebra

Question

Simplify the following expressions. $$ \frac{\sqrt{8}}{\sqrt{3 x}+\sqrt{2 x}} $$

Step-by-Step Solution

Answer

Rewrite the expression with exponents and simplify: $$\frac{\sqrt{8}}{\sqrt{3x}+\sqrt{2x}} = \frac{2^{\frac{3}{2}}}{x^{\frac{1}{2}}(3^{\frac{1}{2}}+2^{\frac{1}{2}})}$$

Step 1: Rewrite square roots as exponents

Rewrite the square roots as exponents, using the property $$\sqrt{n}=n^{\frac{1}{2}}$$: $$\frac{\sqrt{8}}{\sqrt{3x}+\sqrt{2x}}=\frac{8^{\frac{1}{2}}}{(3x)^{\frac{1}{2}}+(2x)^{\frac{1}{2}}}$$

Step 2: Simplify the numerator and denominator individually

Recall that when a power is distributed to a product of numbers, it can be applied to each factor individually: $$8^{\frac{1}{2}}=(2^3)^{\frac{1}{2}}=2^{\frac{3}{2}}$$ $$(3x)^{\frac{1}{2}}=3^{\frac{1}{2}}x^{\frac{1}{2}}$$ $$(2x)^{\frac{1}{2}}=2^{\frac{1}{2}}x^{\frac{1}{2}}$$ The expression now becomes: $$\frac{2^{\frac{3}{2}}}{3^{\frac{1}{2}}x^{\frac{1}{2}}+2^{\frac{1}{2}}x^{\frac{1}{2}}}$$

Step 3: Factor out the common factor in the denominator and simplify the expression

Factor out the common factor $$x^{\frac{1}{2}}$$ in the denominator: $$\frac{2^{\frac{3}{2}}}{x^{\frac{1}{2}}(3^{\frac{1}{2}}+2^{\frac{1}{2}})}$$ The simplified expression is: $$\frac{2^{\frac{3}{2}}}{x^{\frac{1}{2}}(3^{\frac{1}{2}}+2^{\frac{1}{2}})}$$

Key Concepts

Radicals and Exponents

Understanding the relationship between radicals and exponents is crucial when simplifying mathematical expressions. A radical, often represented by a square root symbol, can be expressed as an exponent to allow for simpler manipulation of the terms. For instance, the square root of a number 'n' can be rewritten as an exponent with frac{1}{2}:\[\sqrt{n} = n^{\frac{1}{2}}\]This is known as the 'fractional exponent rule.' It translates the process of finding a root into a more familiar operation of exponential expressions. When you encounter an expression like \[\frac{\sqrt{8}}{\sqrt{3x}+\sqrt{2x}}\], converting each square root to fractional exponent form makes it easier to apply algebraic rules for simplification.
In educational content, this transformation allows students to use the properties of exponents, such as the power of a product and the power of a power, to simplify expressions. This approach can transform a complex-looking radical expression into a more approachable algebraic form.

Factoring Common Factors

Factoring out common factors in an expression is a method used to simplify algebraic expressions by reducing them to their simplest form. It involves identifying and extracting the highest common factor among different terms. This is particularly important when the terms are under a radical, as it can significantly simplify the expression. For example:\[\frac{2^{\frac{3}{2}}}{3^{\frac{1}{2}}x^{\frac{1}{2}} + 2^{\frac{1}{2}}x^{\frac{1}{2}}}\]involves factoring out the common factor of \(x^{\frac{1}{2}}\) from the denominator:\[\frac{2^{\frac{3}{2}}}{x^{\frac{1}{2}}(3^{\frac{1}{2}} + 2^{\frac{1}{2}})}\]
The process of factoring simplifies the overall structure of the expression and often reveals a more direct path to the solution. When explaining this to students, emphasize the value of checking for common factors in order to make seemingly complex problems much more manageable.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in mathematics, aiming to rewrite expressions in a more manageable form. Simplification can involve a combination of expanding, factoring, cancelling, and combining like terms. To simplify an expression, we often follow a certain order:

• Employ the distributive law to expand expressions.
• Combine like terms by addition or subtraction.
• Factor expressions to reveal common factors and simplify.
• Reduce fractions by finding and cancelling common factors in the numerator and denominator.

In the context of square roots and radicals, simplification often involves turning the roots into exponents and then performing algebraic operations to streamline the expression. Simplifying the given exercise, we converted square roots to fractional exponents, factored out a common factor from the denominator, and were left with an expression that is easier to handle. Throughout this process, it is imperative to exercise patience and attention to detail; this ensures that every step is clear and that the final answer is indeed the simplest form of the original expression.