Problem 22 - A Graphical Approach to College Algebra

Question

Simplify each expression. Assume that all variables represent positive real numbers. $$\left(36 r^{6}\right)^{1 / 2}$$

Step-by-Step Solution

Answer

The simplified expression is \(6r^3\).

Step 1: Understand the Expression

We have the expression \((36 r^{6})^{1/2}\). This expression involves taking the square root of a product. Our task is to simplify this expression as much as possible.

Step 2: Apply the Power Rule for Exponents

The power rule for exponents states that \((a^m)^n = a^{m imes n}\). Applying this rule, we can rewrite \((36 r^{6})^{1/2}\) as two separate components: \((36)^{1/2}\) and \((r^{6})^{1/2}\).

Step 3: Simplify the Square Root of the Constant

First, simplify \((36)^{1/2}\). The square root of 36 is 6 since \(6^2 = 36\). Thus, \((36)^{1/2} = 6\).

Step 4: Simplify the Square Root of the Variable

Now simplify \((r^6)^{1/2}\). Using the power rule, we obtain \(r^{6 imes 1/2} = r^3\).

Step 5: Combine the Simplified Expressions

Combine the results from Step 3 and Step 4 to get the simplified expression. You will then have \(6 \times r^3 = 6r^3\).

Key Concepts

Square Root

The concept of a square root is fundamental in mathematics, especially when dealing with quadratic expressions and functions. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 36 is 6, because

• \(6 \times 6 = 36\)

In expressions, the square root is often represented with the radical symbol \( \sqrt{} \) or exponent notation \((^{1/2})\).

• Example: \(\sqrt{36} = 36^{1/2}\)

When simplifying expressions, it's important to break down numbers getting the simplest form. This is crucial for understanding and simplifying larger and more complex mathematical problems.

Power Rule for Exponents

The power rule for exponents is an essential tool in algebra that helps simplify expressions involving powers. It states that

• \((a^m)^n = a^{m \times n}\)

This rule allows you to multiply the exponents when raising a power to another power. In our example, we simplify by separating

• \((36 r^6)^{1/2} = (36)^{1/2} \times (r^6)^{1/2}\)

and then apply the power rule to the variable:

• \((r^6)^{1/2} = r^{6 \times 1/2} = r^3\)

Mastering this rule can greatly simplify complex algebraic expressions and is a foundational skill in algebra.

Simplifying Variables

Simplifying variables involves reducing expressions to their most basic form. This is important for making problems more understandable and solving them efficiently. In our example, we portioned out the expression with variables:

• \( (r^6)^{1/2} = r^{3} \)

This simplification occurs because when applying the power rule, the expression becomes much simpler. Recognizing patterns where variables can be simplified is crucial. For any expression with exponents, always:

• Identify the variable part
• Apply exponent rules like multiplying or dividing powers

This process not only makes calculations easier but also prepares you for more advanced topics in mathematics.