Problem 38 - College Algebra

Question

For the following exercises, simplify each expression. $$ \left(144 p^{2} q^{6}\right)^{\frac{1}{2}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 12pq^3 \).

Step 1: Understand the Expression

The expression given is \( \left(144 p^{2} q^{6}\right)^{\frac{1}{2}} \). This is a radical expression and needs to be simplified using the rules of exponents and square roots.

Step 2: Apply the Square Root to Each Term

Distribute the square root over each part of the expression inside the parenthesis: \[\sqrt{144} \times (p^2)^{\frac{1}{2}} \times (q^6)^{\frac{1}{2}}.\] This allows us to handle the constants and each variable separately.

Step 3: Simplify the Constant

The square root of 144 is 12 because \( 12 \times 12 = 144 \). So \( \sqrt{144} = 12 \).

Step 4: Simplify the Power of p

Using the rule \((a^m)^{n} = a^{m \cdot n}\), compute \((p^2)^{\frac{1}{2}} = p^{2 \cdot \frac{1}{2}} = p^{1} = p\).

Step 5: Simplify the Power of q

Similarly, compute \( (q^6)^{\frac{1}{2}} = q^{6 \cdot \frac{1}{2}} = q^{3} \).

Step 6: Combine the Simplified Terms

Combine all the simplified parts from the previous steps to get the final simplified expression: \( 12pq^3 \).

Key Concepts

Rules of Exponents

The rules of exponents are essential tools in algebra, helping us to simplify expressions effectively. When we talk about exponents, we refer to numbers that tell us how many times to multiply a base by itself. For example, in the expression \( a^n \), \( a \) is the base, and \( n \) is the exponent, indicating that \( a \) is used as a factor \( n \) times.

Some fundamental rules of exponents include:

• The Product Rule: \( a^m \times a^n = a^{m+n} \)
• The Power Rule: \( (a^m)^n = a^{m\cdot n} \)
• The Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \)
• Zero Exponent Rule: \( a^0 = 1 \) (where \( a eq 0 \))

Understanding these rules allows you to break down complex expressions like \((p^2)^{\frac{1}{2}}\) and reason through their simplification, such as \(p^{2 \cdot \frac{1}{2}} = p^1 = p\). The Power Rule is particularly powerful, allowing us to manage nested exponents with ease.

Radical Expressions

Radical expressions involve roots, and the most commonly encountered is the square root. In a radical expression such as \( \sqrt{144} \), the number 144 inside the radical symbol is known as the radicand. The goal with radicals is often to simplify them by breaking down the radicand into perfect squares.

The symbol \( \sqrt{} \) typically represents the principal (or positive) square root. To simplify radical expressions, you often identify a perfect square within the radicand. If the radicand is not a perfect square but can be factored to have one, simplify it accordingly. In our exercise, 144 is a perfect square because \( 12 \times 12 = 144 \), so \( \sqrt{144} = 12 \).

Meanwhile, for components like \( (q^6)^{\frac{1}{2}} \), converting the exponent \( \frac{1}{2} \) into a radical form helps understand that you are looking for the square root of \( q^6 \), leading to simplified expressions like \( q^{6 \cdot \frac{1}{2}} = q^3 \).

Square Root

The square root is a special case of an exponent where the power is \( \frac{1}{2} \). Notation like \( \sqrt{x} \) means you're looking for a value that, when multiplied by itself, gives \( x \). Knowing the square roots of numbers can greatly simplify evaluations and calculations.

To simplify expressions involving square roots, it is crucial to spot perfect squares quickly. Identifying that 144 is a perfect square, because it is \( 12^2 \), ensures you can simplify \( \sqrt{144} \) directly to 12.

When variables are involved, you apply the square root by using exponents. With \( p^2 \), taking the square root is equivalent to raising it to the power of \( \frac{1}{2} \), yielding \( p \). Similarly, for \( q^6 \), \( (q^6)^{\frac{1}{2}} = q^3 \). These steps help in breaking down seemingly complex expressions into manageable pieces.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operation symbols, organized to show a mathematical relationship. When simplifying algebraic expressions, the aim is often to rewrite them in a more concise form without changing their value.

In our example, the expression \((144 p^2 q^6)^{\frac{1}{2}}\) involves numbers and variables that can represent a number of real-world quantities. Simplifying such expressions often means using rules of exponents and manipulating radicals to achieve a simplified form, such as \( 12pq^3 \).

Breaking down the expression:

• Recognize the components: numbers (144), and variables (\( p^2 \), \( q^6 \)).
• Apply mathematical rules to each part: rule of square roots and exponents.
• Combine simplified components to construct the final expression.

By following these steps, you ensure clarity and accuracy in simplifying algebraic expressions, which is vital in both academic and real-world applications.