Problem 68 - Elementary Algebra

Question

Simplify. $$ \sqrt{q^{5}} $$

Step-by-Step Solution

Answer

The simplified form is \(q^{2} \times \sqrt{q}\)

Step 1: Understand the Expression

Recognize that you need to simplify the square root of the expression. Here, the given expression is \(\begin{displaymath}\begin{aligned}\begin{aligned} \sqrt{q^{5}} \end{aligned}\begin{aligned} \end{aligned} \end{aligned} \end{displaymath}\).\

Step 2: Use the Property of Radicals

Use the property that \(\begin{displaymath}\begin{aligned}\begin{aligned} \sqrt{a^{b}} = a^{\frac{b}{2}} \end{aligned}\begin{aligned} \end{aligned} \end{aligned} \end{displaymath} \). Apply this to the expression to get \(q^{\frac{5}{2}}\).\

Step 3: Separate the Expression

Further simplify by separating powers to write the expression as \( \begin{displaymath}\begin{aligned}\begin{aligned} \sqrt{q^{4} \times q^{1}} = \sqrt{q^{4}} \times \sqrt{q} \end{aligned} \end{aligned} \end{displaymath} \).\

Step 4: Simplify Each Term

Simplify each radical term separately, \( \begin{displaymath}\begin{aligned}\begin{aligned} \sqrt{q^{4}} = q^{2} \end{aligned} \end{aligned} \end{displaymath} \) and \( \begin{displaymath}\begin{aligned}\begin{aligned} \sqrt{q} \end{aligned} \end{aligned} \end{displaymath} \). Combining these, the expression becomes \( \begin{displaymath}\begin{aligned}\begin{aligned} q^{2} \times \sqrt{q} \end{aligned} \end{aligned} \end{displaymath}\).\

Key Concepts

Square Root

To understand the concept of simplifying radicals, it's crucial to grasp what a square root is. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).

In algebra, we use similar principles but with variables. The square root of a variable raised to an exponent can be simplified using properties of exponents and radicals. For instance, simplifying \(\sqrt{q^5}\) involves reducing the expression under the radical symbol.

Exponents

Exponents indicate how many times to multiply a number by itself. In the expression \(q^5\), the exponent is 5, meaning \(q\) is multiplied by itself five times: \(q \times q \times q \times q \times q\).

Understanding how exponents work is essential for simplifying radicals. When dealing with square roots, you use the property \(\sqrt{a^b} = a^{\frac{b}{2}}\). This rule helps convert the radical expression into an exponential form, which is often easier to handle mathematically.

Radical Simplification

To simplify radicals, especially those involving exponents, use a step-by-step approach:

• Rewrite the expression in exponential form using the rule \(\sqrt{a^b} = a^{\frac{b}{2}}\).
• For \(\sqrt{q^5}\), apply this rule to get \(q^{\frac{5}{2}}\).
• Now, separate the powers to make the simplification easier: \(q^{\frac{5}{2}} = q^{2+\frac{1}{2}} = q^2 \times q^{0.5}\).
• Finally, rewrite in radical form: \(q^2 \times \sqrt{q}\).

By breaking down the steps, you make it easier to follow the process and ensure accuracy in simplifying radical expressions.