Problem 38 - College Algebra

Question

Simplify each expression. $$\left(144 p^{2} q^{6}\right)^{\frac{1}{2}}$$

Step-by-Step Solution

Answer

The simplified expression is \(12pq^3\).

Step 1: Apply the Power of a Power Rule

The expression is \( (144 p^2 q^6)^{\frac{1}{2}} \). According to the power of a power rule, we need to apply the exponent \( \frac{1}{2} \) to each term inside the parenthesis separately. This rule states that \( (a^m)^n = a^{m \cdot n} \).

Step 2: Simplify Each Component Separately

Apply the \( \frac{1}{2} \) exponent:- For \( 144 \), calculate \( 144^{\frac{1}{2}} \). Since \( 144 \) is a perfect square (\( 12^2 \)), this simplifies to \( 12 \).- For \( p^2 \), calculate \( (p^2)^{\frac{1}{2}} \), which gives \( p^{2 \cdot \frac{1}{2}} = p^1 = p \).- For \( q^6 \), calculate \( (q^6)^{\frac{1}{2}} \), which gives \( q^{6 \cdot \frac{1}{2}} = q^3 \).

Step 3: Combine All Simplified Elements

The expression is now simplified to the product of the simplified components: \[ 12 \cdot p \cdot q^3 \] Thus, the simplified form of the expression is \( 12p q^3 \).

Key Concepts

Power of a Power Rule

The Power of a Power Rule is a fundamental concept in algebra that makes simplifying expressions with nested exponents much easier. When you have something like \((a^m)^n\), this rule tells us that we can multiply the exponents, transforming it into \(a^{m \cdot n}\). This simplification is handy when dealing with complex algebraic expressions.

Consider, for example, the expression \((p^2)^{\frac{1}{2}}\). By applying the Power of a Power Rule, we multiply the exponents: \(2 \times \frac{1}{2} = 1\). This simplifies the expression to \(p^1\), or simply \(p\).

This rule allows us to work through expressions systematically and efficiently, ensuring that each component of a formula is simplified correctly.

Perfect Squares

Perfect squares are numbers that can be expressed as the square of an integer. In other words, a perfect square results from multiplying an integer by itself. Recognizing perfect squares can simplify the process of finding square roots.

For example, in the expression \((144)^{\frac{1}{2}}\), we notice that 144 is a perfect square because it equals \(12^2\). Knowing this, we can easily find that \(144^{\frac{1}{2}}\) simplifies to 12.

Recognizing perfect squares allows for the quick simplification of expressions, particularly when dealing with roots and fractional exponents. It's a useful technique that can save you a lot of time and effort in math.

Exponentiation

Exponentiation is the process of raising a number to a power. It involves expressing a number in terms of another number, known as the base, and the power or exponent. This notation, \(a^n\), means multiplying the base \(a\) by itself \(n\) times.

In algebra, exponentiation helps handle large numbers efficiently and forms the basis for simplifying expressions. For instance, in \(144 p^2 q^6\), each variable and number is subjected to an exponent. Using exponentiation rules, you simplify the expression by applying the exponent to each element.

Exponentiation is integral to algebra. Whether you're multiplying numbers or simplifying variables, understanding how to manage exponents is key to successfully solving mathematical problems.