Problem 63 - Algebra 1

Question

Using the fact that \(x^{1 / 2}=\sqrt{x}\), rewrite in simplest radical form. $$x^{1 / 2} \cdot 4 \sqrt{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(4 \sqrt{2x}\)

Step 1: Substitute the power

Since \(x^{1 / 2}=\sqrt{x}\), replace \(x^{1 / 2}\) in \(x^{1 / 2} \cdot 4 \sqrt{2}\) with \(\sqrt{x}\). Therefore, expression becomes \(\sqrt{x} \cdot 4 \sqrt{2}\)

Step 2: Simplify the expression

Rewrite the expression by multiplying the two radical terms, \(\sqrt{x}\) and \(\sqrt{2}\), together. According to the multiplication rule for radicals \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\). So, the expression becomes \(4 \sqrt{2x}\)

Step 3: Final simplification

No further simplification is possible, so the final expression is \(4 \sqrt{2x}\).

Key Concepts

Simplest Radical Form

When dealing with radicals, one of the key skills is simplifying them to their simplest form. This means expressing the radical in the most concise way without changing its value.
The simplest radical form is achieved by ensuring that:

• There are no perfect square factors left in the radicand (the number under the square root).
• Any coefficients are kept outside the radical.

For example, when you have an expression like \(\sqrt{50}\), you should break it down into its prime factors. Notice that 50 can be expressed as \(25 \times 2\). Since 25 is a perfect square, we can simplify the radical to \(5 \sqrt{2}\).

In our exercise, when transforming \(x^{1/2}\) to \(\sqrt{x}\), we are beginning this process of simplifying as we prepare the expression for further operations.

Multiplication of Radicals

Multiplying radicals can be smooth if you follow the correct rules. To multiply radicals together effectively, you need to understand some straightforward properties:

• When you multiply a number outside the radical by another outside number, you simply multiply them as normal numbers.
• When you multiply radicals together, you use the property \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\).

In the original exercise, the expression \(\sqrt{x} \cdot 4\sqrt{2}\) uses this multiplication rule. The term \(4\) remains outside as a coefficient.
Inside the radicals, \(\sqrt{x} \cdot \sqrt{2}\), become \(\sqrt{2x}\) once combined. This results in the expression \(4\sqrt{2x}\).

By following these straightforward steps and rules, multiplying radicals doesn't have to be difficult. Always remember to merge the radicands first and keep outer numbers separate until the end.

Exponents and Radicals

Understanding the relationship between exponents and radicals can make a big difference in simplifying expressions. Fundamentally, a radical such as a square root is just another way to express an exponent. Specifically, \(x^{1/2}\) is the same as \(\sqrt{x}\).

This interchangeability is crucial for transforming expressions with fractional exponents into radical form, making them simpler to understand and manipulate. For instance:

• \(x^{1/2}\) is \(\sqrt{x}\).
• \(x^{1/3}\) would be \(\sqrt[3]{x}\).

Any time you encounter an exponent that is a fraction, you can think of it as a radical. This simplifies working with complex algebraic expressions. It provides a clearer perspective on tasks such as simplifying, multiplying, or factoring expressions.

In our problem, converting \(x^{1/2}\) to \(\sqrt{x}\) made combining with the \(4\sqrt{2}\) term possible, demonstrating how exponents and radicals interact in algebra.