Problem 24 - Elementary and Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt{75 b^{8} c} $$

Step-by-Step Solution

Answer

The simplified expression is \(5b^4 \sqrt{3c}\).

Step 1: Identify Perfect Squares

First, note that the radical expression we have is \( \sqrt{75 b^8 c} \). We need to identify any perfect square factors in each part of the expression. For numbers, look at factors of 75: \(75 = 25 \times 3 = 5^2 \times 3\), so 25 is a perfect square. For variables, \(b^8\) is \((b^4)^2\), which is a perfect square.

Step 2: Rewrite the Radical

Using our understanding from Step 1, we can rewrite the expression: \[ \sqrt{75 b^8 c} = \sqrt{5^2 \times 3 \times (b^4)^2 \times c}. \]

Step 3: Simplify the Radical

Extract the perfect squares from under the radical, which are \(5^2\) and \((b^4)^2\):\[= \sqrt{5^2} \times \sqrt{(b^4)^2} \times \sqrt{3c} = 5 \times b^4 \times \sqrt{3c}.\]

Step 4: Write the Final Simplified Expression

Thus, the simplified form of the radical expression is:\[ 5b^4 \sqrt{3c}. \]

Key Concepts

Perfect Square Factorization

When simplifying radical expressions, a great technique is to use perfect square factorization. Perfect squares are numbers or variables raised to the power of 2, such as 4, 9, and 16, or variables like \(a^2\) and \(b^4\). These are useful because their square roots are whole numbers or simpler expressions.
To factor a number or variable into perfect squares:

• Look for numbers that can be expressed as a product of two identical factors, such as \(25 = 5^2\).
• For variables, check their exponents. Even exponents signify perfect squares. So, \(b^8\) becomes \((b^4)^2\).

In our radical expression \(\sqrt{75 b^8 c}\), we identified \(25\) and \(b^8\) as perfect squares. This was a crucial first step in simplifying the expression.

Simplifying Square Roots

Once you've identified the perfect squares, you can simplify the square roots, which means finding their square roots and taking them out of the radical. Here's how it works:

• For numbers that are perfect squares like \(5^2\), the square root is simply the number itself, in this case, \(5\).
• For variables like \((b^4)^2\), the square root is \(b^4\), because the square root cancels out the exponent of 2.

The important thing at this stage is to extract these perfect squares from the root to simplify the expression. So, from \(\sqrt{75 b^8 c} = \sqrt{5^2 \times 3 \times (b^4)^2 \times c}\), we can simplify it to \(5b^4 \sqrt{3c}\). Notice how the leftover product \(3c\) remains inside the square root because it isn't a perfect square.

Algebraic Expressions

Algebraic expressions present a combination of numbers, variables, and operations. When dealing with radical expressions like \(\sqrt{75 b^8 c}\), understanding these components helps in rearranging and simplifying.
In algebra:

• Variables represent numbers and can be raised to powers, which helps in factoring.
• Constant terms combine with variables to form parts of expressions that can be simplified according to algebraic rules.

The algebraic manipulation in simplifying radicals includes recognizing and extracting perfect squares and simplifying leftover terms within the radical. The result is a more refined algebraic expression: \(5b^4 \sqrt{3c}\), which is simpler and easier to interpret.