Problem 27 - Algebra for College Students

Question

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt{64 x^{3} y^{7}}\)

Step-by-Step Solution

Answer

The simplest radical form is \(8xy^3 \sqrt{xy}\).

Step 1: Split the Expression

First, we will determine the separate components inside the square root: \(64\), \(x^3\), and \(y^7\). We will express these separately under the square root: \(\sqrt{64}\), \(\sqrt{x^3}\), and \(\sqrt{y^7}\).

Step 2: Simplify \(\sqrt{64}\)

Calculate the square root of 64. Since \(64 = 8^2\), the square root \(\sqrt{64}\) simplifies to \(8\).

Step 3: Simplify \(\sqrt{x^3}\)

For \(x^3\), break it as \(x^2 \cdot x^1\). The square root \(\sqrt{x^2 \cdot x^1}\) simplifies to \(x \sqrt{x}\) since the square root of \(x^2\) is \(x\).

Step 4: Simplify \(\sqrt{y^7}\)

For \(y^7\), break it into components of \(y^6 \cdot y^1\). The square root of \(y^6\) is \(y^3\), and we are left with \(\sqrt{y}\). Thus, \(\sqrt{y^7} = y^3 \sqrt{y}\).

Step 5: Combine the Simplified Terms

Finally, combine all simplified components: \(8\), \(x \sqrt{x}\), and \(y^3 \sqrt{y}\). This yields: \(8xy^3 \sqrt{xy}\).

Key Concepts

Square Root Simplification

Simplifying square roots is like finding the easiest way to express an expression under the radical symbol. The goal is to make this expression as simple as possible, which is sometimes called the "simplest radical form." Here’s how we break it down:

• Identify perfect square factors. These are numbers whose square root can be determined exactly, like 4, 9, 16, 25, etc.
• Any perfect squares can be "moved out" from under the square root symbol.

For example, if you have \(\sqrt{64}\), since 64 is a perfect square (8 squared), it simplifies directly to 8.

Radical Expressions

A radical expression contains a square root, cube root, or any higher order root. These expressions often involve variables and numbers under a radical symbol. Understanding how to manipulate parts of these expressions helps in simplifying them and in algebraic operations later on.

Examining \(\sqrt{x^3}\), we recognize that you can simplify this by breaking it into parts: \(\sqrt{x^2 \cdot x}\). The term \(x^2\) is a perfect square, so \(\sqrt{x^2}\) becomes \(x\), leaving us with \(x \sqrt{x}\) as the simplified form. This way, you ensure part of your radical expression has been moved out from under the square root.

Algebraic Manipulation

Algebraic manipulation allows us to modify and simplify expressions without changing their value. It’s a core skill for dealing with radical expressions. Let's go step-by-step.

We begin with separating components as seen with \(\sqrt{y^7}\). This can be split into \(y^6\cdot y\). Since \(y^6\) is \(\left(y^3\right)^2\), its square root is \(y^3\). After taking the square root, you get \(y^3\sqrt{y}\), which shows how manipulation helps to simplify completely.

• Break down expressions to their simplest parts.
• Recombine these parts in a way that reflects the original expression but in a simpler form.

Combining everything provides \(8xy^3 \sqrt{xy}\), which is a result of applying square root simplification, understanding radical expressions, and employing algebraic manipulation.