Problem 44 - Intermediate Algebra

Question

Simplify each expression. Assume that all variables are unrestricted and use absolute value symbols when necessary. See Example 2. $$ \sqrt{9 y^{6}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 3|y^3| \).

Step 1: Analyze the Expression

The expression to simplify is \( \sqrt{9y^6} \). We will first break this down—the square root of a product can be separated into the product of the square roots. Our expression can be considered as \( \sqrt{9} \times \sqrt{y^6} \).

Step 2: Simplify the Numeric Part

Calculate \( \sqrt{9} \). This is straightforward because \( 9 \text{ is a perfect square} \), so \( \sqrt{9} = 3 \).

Step 3: Simplify the Exponential Part

Now we address \( \sqrt{y^6} \), which is the square root of an exponential expression. The square root rule for exponents tells us that \( \sqrt{y^6} = y^{6/2} = y^3 \).

Step 4: Consider Absolute Values

Since the original variable \( y \) is unrestricted, we must apply absolute value because \( y^3 \text{ is raised to an even exponent before the square root} \). Therefore, \( \sqrt{y^6} = |y^3| \).

Step 5: Combine the Simplified Parts

By combining both independently simplified parts, we get \( \sqrt{9} \times \sqrt{y^6} = 3 \times |y^3| = 3|y^3| \). Thus, the simplified expression is \( 3|y^3| \).

Key Concepts

Understanding Square Root Properties

Square roots are fundamental in algebra and simplifying expressions often requires their properties. The square root of a number or an algebraic expression is a value that, when multiplied by itself, gives the original number or expression. Understanding a few key properties makes it easier to simplify complex expressions, especially when dealing with products and quotients.

One useful property is that the square root of a product can be separated into the product of square roots. For example, \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \). This allows us to simplify expressions like \( \sqrt{9y^6} \) by separating \( 9 \) and\( y^6 \).

Another property involves perfect squares. If a number inside the square root is a perfect square, it simplifies directly to its integer root. In our example, \( \sqrt{9} = 3 \) because 9 is a perfect square.

Exploring the Exponent Rules

Exponent rules are instrumental in simplifying expressions involving powers. They help break down complex expressions, making them easier to handle. Let's dive into some key rules using our example, \( \sqrt{y^6} \).

The square root of a power expression can be simplified using the rule: \( \sqrt{a^m} = a^{m/2} \). This allows us to convert \( \sqrt{y^6} \) into a more manageable form \( y^{6/2} = y^3 \).

Further rules include the product of powers rule, which states \( a^m \times a^n = a^{m+n} \), and the power of a power rule, which is \((a^m)^n = a^{m \cdot n} \). These rules together provide a powerful toolkit for algebraic simplification.

The Role of Absolute Value

Absolute value acts as a critical component when working with expressions involving square roots and exponents, especially when variables can be unrestricted. The absolute value of a number or expression is its non-negative value, regardless of its sign.

In expressions like \( \sqrt{y^6} = y^3 \), since \( y \) can be any real number, we wrap it in absolute value to ensure we account for all cases, yielding \( |y^3| \). This is essential because squaring a negative number results in a positive value. Therefore, when reversed, using the square root, it's necessary to ensure the result fits the original conditions.

This approach ensures the solution remains valid across all possible values for the variables, maintaining accuracy and completeness in algebraic simplification.