Problem 32 - Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt{147 a^{5}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 7a^2 \sqrt{3a} \).

Step 1: Identify Perfect Squares

Examine the radicand, in this case, 147, to find any perfect square factors it may have. Note that 147 can be factored into 49 and 3, where 49 is a perfect square (49 = 7^2).

Step 2: Write Radicand as Product of Perfect Square and Other Factor

Express 147 as a product of the perfect square and another factor: 147 = 49 \times 3. Therefore, we have \( \sqrt{147a^5} = \sqrt{49 \times 3 \times a^5} \).

Step 3: Simplify Using Square Root Property

Use the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). Apply this to \( \sqrt{49 \times 3 \times a^5} \), giving us \( \sqrt{49} \times \sqrt{3} \times \sqrt{a^5} \).

Step 4: Simplify Each Factor

Calculate each square root: \( \sqrt{49} = 7 \) and for \( a^5 \), rewrite it as \( a^4 \times a \). This gives us \( \sqrt{a^5} = \sqrt{a^4 \times a} = \sqrt{a^4} \times \sqrt{a} = a^2 \times \sqrt{a} \), because \( \sqrt{a^4} = a^2 \).

Step 5: Combine the Simplified Terms

The expression simplifies to \( 7 \times a^2 \times \sqrt{3} \times \sqrt{a} \). Combine the radicals: \( 7a^2 \sqrt{3a} \).

Key Concepts

Perfect Square

A **perfect square** is an integer that is the square of another integer. This means that when you multiply an integer by itself, you get a perfect square. For example, 1, 4, 9, 16, and 25 are all perfect squares because they are squared numbers of 1, 2, 3, 4, and 5, respectively. Recognizing these numbers is crucial when simplifying radicals because perfect squares can easily be taken out of a square root.

For instance, in the problem we tackled, the number 49 is a perfect square (\(49 = 7^2\)). Identifying this allows us to simplify \(\sqrt{49}\) to 7, making the simplification process much easier.

• Perfect squares help in breaking down larger numbers under a square root.
• They make the calculation simpler by converting radicands into integers.

Spotting perfect squares can save time and effort in mathematical operations, especially in simplifying radical expressions.

Radical Expressions

**Radical expressions** include square roots, cube roots, and any other expressions containing roots. The most common radical expression involves the square root, symbolized by the radical sign (√).

When simplifying radical expressions, our goal is to express them in their simplest form by reducing the expression under the root as much as possible. This involves:

• Identifying and factoring out perfect squares.
• Breaking down higher powers of variables (like \(a^5\)) into manageable parts, often separating them into even exponents to simplify.

In our example, we dealt with the radical expression \(\sqrt{147a^5}\). By factorizing, we expressed 147 as 49 and 3, leading to a simpler form. Similarly, the term \(a^5\) was expressed as \(a^4 \times a\), allowing for simpler square root extraction.

Square Root Property

The **square root property** is a crucial mathematical property used in simplifying radicals. It states:\[\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\]This property allows us to break down complex radical expressions into simpler parts that can be individually simplified and then recombined.

In practical application, you need to:

• Identify multiplicative components under the square root that themselves can be broken down.
• Apply the square root property to separate and simplify each factor.

For the given problem, we used this property to simplify:\[\sqrt{49 \times 3 \times a^5} = \sqrt{49} \times \sqrt{3} \times \sqrt{a^5}\]By applying the square root property, each factor was tackled individually, leading to a final simplified expression of \(7a^2\sqrt{3a}\).