Problem 47 - College Algebra with Modeling and Visualization

Question

Simplify the radical expression by factoring out the largest perfect nth power. Assume that all variables are positive. $$ \sqrt{12 a^{2} b^{5}} $$

Step-by-Step Solution

Answer

The simplified expression is \(2ab^2\sqrt{3b}\).

Step 1: Identify Perfect Squares

The expression under the square root is \(12a^2b^5\). Begin by identifying perfect square factors in each part of the expression. For 12, the perfect square factor is 4 (since \(4 \times 3 = 12\)), for \(a^2\), it is already a perfect square, and for \(b^5\), the perfect square factor is \(b^4\) (since \((b^2)^2 = b^4\)).

Step 2: Rewrite the Expression

Rewrite the expression using the identified perfect squares: \(\sqrt{4 \cdot 3 \cdot a^2 \cdot b^4 \cdot b}\).

Step 3: Simplify the Perfect Squares Out of the Radical

Take the square roots of the perfect square factors out of the radical: \(\sqrt{4} \cdot \sqrt{a^2} \cdot \sqrt{b^4} \cdot \sqrt{3b}\). This simplifies to \(2 \cdot a \cdot b^2\) times the remaining \(\sqrt{3b}\).

Step 4: Combine the Results

Combine the factors outside of the radical: The simplified form of the expression is \(2ab^2\sqrt{3b}\).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots and cube roots. When simplifying these expressions, we aim to make them as simple as possible, sometimes by finding perfect powers. In this task, we deal with a square root, represented by the radical symbol \( \sqrt{} \). This means we're looking for what number multiplied by itself gives us the expression under the root.
The expression inside the root is called the radicand, which in our example is \(12a^2b^5\). Simplifying radical expressions can involve breaking down the radicand into parts, typically looking for perfect squares if the root is a square root. This is because perfect squares are easy to deal with: their square roots are whole numbers, which help in simplifying the expression overall.
To start, factorize the numbers and variables in the radicand to isolate the perfect squares and simplify effectively.

Perfect Squares

A perfect square is a number or expression that can be written as the square of an integer. For example, 4 and 9 are perfect squares because \(2^2 = 4\) and \(3^2 = 9\). Recognizing perfect squares in algebra helps us simplify expressions, especially those under a square root. Here are some examples of common perfect squares:

• 4, because \(2 \times 2 = 4\)
• 16, because \(4 \times 4 = 16\)
• \(a^2\), because \(a \times a = a^2\)

In the expression \(\sqrt{12a^2b^5}\), discerning which parts are perfect squares directly influences how we simplify it. For instance, \(12\) can be broken into \(4 \times 3\), with 4 being a perfect square. In the term \(b^5\), \(b^4\) is a perfect square as it is equal to \((b^2)^2\). Recognizing and extracting these will simplify the radical greatly.

Simplifying Expressions

Simplifying expressions is all about transforming complicated math expressions into their simplest form. Here, it involves breaking down the expression inside the square root by identifying perfect square factors:

• For numbers like 12, it can be broken down into \(4 \times 3\), where 4 is a perfect square.
• For variables, \(a^2\) and \(b^4\) are perfect squares because their powers are divisible by 2.

The idea of simplification involves extracting these perfect squares from the radicand and taking them outside the root. Once they are outside, their influence within diminishes, resulting in a simpler, cleaner expression like \(2ab^2\sqrt{3b}\). This involves taking the square root of perfect squares and multiplying them with any other remaining terms either inside or outside the root.

Factoring

Factoring is fundamental in simplifying algebraic expressions, particularly when dealing with radicals. It involves breaking down a composite number or expression into its factorable components:

• For example, factorizing 12 to \(4 \times 3\), leverages 4 as a perfect square.
• Similarly, splitting \(b^5\) to \(b^4 \times b\) helps in identifying \(b^4\) as a perfect square.

In this process, we identify and extract perfect squares, stepping toward simplification. Factoring becomes especially vital in lengthy expressions as it isolates portions we can then manage more easily. Once these are isolated, further simplification involves removing the square roots of these factors, leading to a solution that's neat and efficient.