Problem 62 - Algebra 1

Question

Using the fact that \(x^{1 / 2}=\sqrt{x}\), rewrite in simplest radical form. $$6 x^{1 / 2}$$

Step-by-Step Solution

Answer

The simplest radical form of the expression \(6x^{1 / 2}\) is \(6\sqrt{x}\)

Step 1: Identify the Exponent Equivalent

Recognize that the exponent of \(1 / 2\) in the expression \(x^{1 / 2}\) is equivalent to the square root of \(x\).

Step 2: Apply the Equivalent

Replace the \(x^{1 / 2}\) in the expression \(6x^{1 / 2}\) with \(\sqrt{x}\).

Step 3: Write in Simplest Radical Form

The expression simplifies to \(6\sqrt{x}\)

Key Concepts

Exponent Equivalent

Understanding exponent equivalents is critical for simplifying mathematical expressions. An exponent equivalent refers to different ways of representing the same exponential expression. For instance, the expression involving an exponent of \frac{1}{2}, such as in the term x^{1/2}, is equivalent to the square root of the variable or number. This relationship is foundational and simplifies the process of working with radical expressions. When you see a fractional exponent, especially with a numerator of 1 and a denominator of 2, you can immediately identify it as a square root. It's crucial to recognize these equivalents to rewrite equations in different but equivalent forms for further simplification.

To make this concept clearer, let's consider the term 9^{1/2}. Applying the equivalent, we rewrite it as \( \( \( \sqrt{9} \) \) \) , which we know equals 3. This process is not just a trick but a rule consistent across all real numbers and is vital for understanding and simplifying more complex algebraic equations.

Square Root

The square root, denoted as \( \sqrt{x} \), plays a significant role in mathematics, especially when dealing with quadratic equations and finding the length of a side of a square with a known area. When a number or expression is under a square root sign, it means we are looking for a value, which, when multiplied by itself, gives us the original number or expression. To express the square root in its simplest radical form, the term inside the radical (also known as the radicand) should not have any perfect square factors other than 1.

For example, if we need to simplify \( \sqrt{18} \), we'd break it down to \( \sqrt{9 \cdot 2} \) because 9 is a perfect square. We can further simplify it to \( 3\sqrt{2} \) because \( \sqrt{9} \) equals 3. Finding the square root of a number or variable is a fundamental skill for solving geometric and algebraic problems.

Radical Expressions

Radical expressions contain a number or expression under a root symbol. The most common type of radical expression is one with a square root; however, radicals can also be cubic roots, fourth roots, and so on. Simplifying radical expressions involves combining like terms under the root and factoring out perfect squares (as square roots) or perfect cubes (as cube roots) whenever possible. A simplified radical has no fractions under the root and no radicals in the denominator after rationalization.

Let's take a closer look by simplifying a radical expression such as \( \sqrt{50} + \sqrt{2} \). We factor 50 to get \( \sqrt{25\cdot2} \) which simplifies to \( 5\sqrt{2} \) since \( \sqrt{25} \) is 5. Combining like terms, we get \( 5\sqrt{2} + \sqrt{2} = 6\sqrt{2} \), which is the simplest form of the expression. Recognizing patterns in radical expressions and knowing how to manipulate them through the properties of radicals is essential for mastering algebra.