Problem 20 - Algebra 2 and Trigonometry

Question

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{\frac{1}{2 x}}+\sqrt{\frac{1}{2 x}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 2\sqrt{\frac{1}{2x}} \).

Step 1: Identify the Problem

We need to simplify the expression \( \sqrt{\frac{1}{2x}} + \sqrt{\frac{1}{2x}} \). Both terms are identical square roots.

Step 2: Combine Like Terms

Since both terms in the expression are identical, they can be combined. This is similar to adding like terms: \( a + a = 2a \). Here, \( \sqrt{\frac{1}{2x}} + \sqrt{\frac{1}{2x}} = 2\sqrt{\frac{1}{2x}} \).

Step 3: Simplify the Expression

The expression \( 2\sqrt{\frac{1}{2x}} \) can be simplified by factoring or handling the constant outside of the square root. Here, \( 2 \) is the coefficient, and \( \sqrt{\frac{1}{2x}} \) is the radical. The expression is already in its simplest form unless further simplification of the radicand is possible.

Key Concepts

Square Roots

Square roots are fundamental in algebra and represent a value that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2, because 2 multiplied by 2 equals 4. In our exercise, we have square roots of fractions, which might seem tricky at first. The expression involves \( \sqrt{\frac{1}{2x}} \). When you see a square root like this, you are looking at the process of finding a number which, when multiplied by itself, gives \( \frac{1}{2x} \). Understanding square roots generally involves knowing properties like:

• The square root of a product \( a \times b \) can be expressed as \( \sqrt{a} \times \sqrt{b} \).
• Square roots of fractions can be broken down: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \)

In this specific exercise, we focus on the term inside the radical \( \frac{1}{2x} \), which is already a fraction. Note that any variable inside a square root, like \( x \) here, is non-negative, ensuring we only deal with real numbers. This understanding helps in approaching and simplifying radical expressions.

Combining Like Terms

Combining like terms is a key algebraic skill useful for simplifying expressions. When expressions are simplified, they often need terms that are similar or identical to be combined to form a clearer result. For example, in arithmetic, you wouldn't leave \( 2 + 2 \) uncombined as 4 is the simpler representation. Our exercise posits the expression \( \sqrt{\frac{1}{2x}} + \sqrt{\frac{1}{2x}} \). Here, both terms are exactly the same. You might recognize this type of problem from elementary mathematics where \( a + a = 2a \).
In this context, it's the same idea applied to radicals:

• When identical square roots are added together, they can be combined into a single term.
• This results in \( 2\sqrt{\frac{1}{2x}} \).

This is a crucial step as it simplifies expressions not just visually, but often make them easier to manipulate in further calculations. Recognizing like terms quickly in algebra is a valuable skill, making problem-solving faster and more efficient.

Rational Expressions

Rational expressions are ratios of two polynomials. When you see them, think fractions in algebraic form. In our exercise, we have \( \sqrt{\frac{1}{2x}} \), which is a bit different as the fraction is under the square root.
Understanding rational expressions involves:

• Recognizing that they have numerators and denominators that are polynomials.
• Ensuring the denominator never equals zero, as division by zero is undefined which is inherently managed for us in most well-posed algebra problems.

In the case of \( \frac{1}{2x} \), the fraction is straightforward: the numerator \( 1 \) and the denominator \( 2x \). The denominator \( 2x \) must never be zero, meaning \( x \) can't be zero.
While rational expressions, as found here, can often be quite simple, they become more complex with higher-order polynomials. However, the principles of treating them like fractions while respecting the rules of operations and not nullifying the denominator remain consistent.