Problem 23 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{a^{5}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\sqrt{a^{5}}\). Answer: \(a^2 \times a^{1/2}\)

Step 1: Identify the largest perfect square that divides the given expression

We need to identify the largest perfect square that divides \(a^5\). In this case, it is \(a^4\), since \(a^4\times a^1 = a^5\).

Step 2: Rewrite the expression

Rewrite the expression as the product of the square root of the largest perfect square and the remaining part: $$ \sqrt{a^5} = \sqrt{a^4 \times a^1} $$

Step 3: Separate the terms under the square root

Separate the terms under the square root into two square root expressions using the property \(\sqrt{ab} = \sqrt{a}\sqrt{b}\): $$ \sqrt{a^4 \times a^1} = \sqrt{a^4}\sqrt{a^1} $$

Step 4: Simplify the square roots

Simplify the square roots: $$ \sqrt{a^4}\sqrt{a^1} = a^2 \times a^{1/2} $$

Step 5: Write the final simplified expression

The simplified expression is: $$ a^2 \times a^{1/2} $$

Key Concepts

Perfect Square

A perfect square is a number that can be expressed as the square of an integer. For example, the number 16 is a perfect square because it equals 4 squared (\(4^2\)). Similarly, algebraic terms like \(a^4\) are perfect squares because they can be written as \((a^2)^2\). Recognizing perfect squares is crucial when simplifying radical expressions, as they provide a means to simplify square roots.

• Perfect squares are integers like 1, 4, 9, 16.
• In algebra, terms like \(a^2, b^4, c^6\) are perfect squares.

When simplifying radicals, look for the largest perfect square that divides the expression. This will help break down complex radicals into simpler parts. For instance, in the expression \(a^5\), the largest perfect square part is \(a^4\), which can be rewritten into two simpler parts. This transformation makes further simplification easier.

Simplifying Radicals

Simplifying radicals involves reducing a radical expression into its simplest form. The process often includes breaking down the expression and finding the largest perfect square factor. For \(\sqrt{a^5}\):

• First, identify the perfect square inside the radical. Here, it's \(a^4\).
• Write the original expression as: \(\sqrt{a^5} = \sqrt{a^4 \times a^1}\).
• Then apply the property of square roots that \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\).

This allows you to handle each part separately and simplifies the radical: \(\sqrt{a^4} \cdot \sqrt{a^1}\). Breaking down the components helps to reach the simplest form in radical expressions.

Square Root Properties

Square root properties are useful rules that help manipulate and simplify expressions under the radical. These properties enable the transformation and combination of square roots in mathematical operations.

• \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\): This allows you to split products under the root.
• \((\sqrt{a})^2 = a\): This means the square root and square are inverse operations.
• \(\sqrt{a^2} = |a|\): Here, the square root of a square is the absolute value of the base.

Using these properties, you can untangle complex roots for easier handling. In the exercise, \(\sqrt{a^5}\) becomes \( \sqrt{a^4} \cdot \sqrt{a^1} \). Further simplifying: \(\sqrt{a^4} = a^2\), leading to the final simple result \(a^2 \cdot a^{1/2}\). Understanding these properties is key in simplifying expressions efficiently and correctly.