Problem 12 - Intermediate Algebra

Question

Simplify. Assume that variables represent positive real numbers. $$ \sqrt{64 y^{20}} $$

Step-by-Step Solution

Answer

The simplified form is \(8y^{10}\).

Step 1: Identify the Square Root Operation

We are tasked with simplifying the expression \( \sqrt{64 y^{20}} \). The provided expression is a square root, meaning we must find the square root of both the numeric coefficient \(64\) and the variable expression \(y^{20}\).

Step 2: Simplify the Square Root of the Coefficient

First, evaluate \( \sqrt{64} \). Since \(64\) is a perfect square, \( \sqrt{64} = 8\). This is because \(8^2 = 64\).

Step 3: Simplify the Square Root of the Variable Expression

Next, simplify \( \sqrt{y^{20}} \). To do this, use the rule \( \sqrt{a^b} = a^{b/2} \). Therefore, \( \sqrt{y^{20}} = y^{20/2} = y^{10} \).

Step 4: Combine Simplified Components

Combine the simplified components from the previous steps. The simplified form of the entire expression is \(8y^{10}\).

Key Concepts

Square Root Rules

Understanding square root rules is essential when simplifying radical expressions like \( \sqrt{64 y^{20}} \). Let's break it down:

• The square root of a product is the product of the square roots: \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \).
• The square root of a quotient is the quotient of the square roots: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).
• For powers, \( \sqrt{a^n} = a^{n/2} \). This is particularly useful for variables.

Applying these rules helps us simplify both the numerical and variable parts of the expression separately, making the process systematic and reliable.

Perfect Squares

Perfect squares are numbers that have an integer as their square root, such as 1, 4, 9, 16, and 64. Recognizing perfect squares is key to simplifying square roots effortlessly. In our expression \( \sqrt{64} \), 64 is a perfect square because \( 8^2 = 64 \). This turns \( \sqrt{64} \) into 8.

• Being aware of perfect squares can save time and reduce errors.
• Familiarize yourself with common perfect squares as they often appear in mathematical problems.

Memorizing perfect squares up to at least 144 can be incredibly beneficial for such simplifications.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves breaking them down into their most basic components without changing their value. Mastering this skill requires an understanding of:

• Combining like terms, especially with variables.
• Applying laws of exponents, such as \( a^m \times a^n = a^{m+n} \) and \( (a^m)^n = a^{mn} \).
• Simplifying radicals by reducing indices with square root rules.

For \( y^{20} \), using \( \sqrt{y^{20}} = y^{20/2} \), we simplify it to \( y^{10} \), combining the terms results in an expression that's quicker to solve and manage: \( 8y^{10} \). Being systematic in your approach ensures clarity and efficiency.