Problem 38 - College Algebra with Corequisite Support

Question

For the following exercises, 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: Recognize the Power Rule

The given expression is \( \left(144 p^{2} q^{6}\right)^{\frac{1}{2}} \). The expression as a whole is raised to the power of \( \frac{1}{2} \), which indicates that we need to take the square root of each component inside the parentheses.

Step 2: Apply the Power to Each Component

Apply the power of \( \frac{1}{2} \) to all parts inside the parentheses: \( \left(144\right)^{\frac{1}{2}} \), \( \left(p^{2}\right)^{\frac{1}{2}} \), and \( \left(q^{6}\right)^{\frac{1}{2}} \). According to the power of a power rule, \( \left(a^{m}\right)^{n} = a^{m \cdot n} \).

Step 3: Simplify the Numerical Component

Calculate the square root of 144: \( 144^{\frac{1}{2}} = 12 \), because \( 12 \times 12 = 144 \).

Step 4: Simplify the Variable with Exponent

Simplify \( \left(p^{2}\right)^{\frac{1}{2}} \) to \( p^{1} \) or simply \( p \) because \( 2 \cdot \frac{1}{2} = 1 \).

Step 5: Simplify the Second Variable's Exponent

Simplify \( \left(q^{6}\right)^{\frac{1}{2}} \) to \( q^{3} \) because \( 6 \cdot \frac{1}{2} = 3 \).

Step 6: Combine the Simplified Components

Combine the simplified components: \( 12 \), \( p \), and \( q^{3} \), resulting in the final expression \( 12pq^{3} \).

Key Concepts

Power Rule

The power rule is crucial when you are simplifying expressions that involve exponents. It helps you understand how to manage powers that are being raised by another power. The rule states that

• If you have an expression of the form \((a^{m})^{n}\) then it can be simplified using the relation \(a^{m \cdot n}\)
• This means you multiply the exponents together.

In our exercise, we applied the power rule when simplifying \(p^{2}\) and \(q^{6}\) by raising them to the power of \(\frac{1}{2}\). This broke down to multiplying the exponents, simplifying the process significantly.

Square Root

A square root is a special fractional exponent represented by raising a number to the \(\frac{1}{2}\) power. It's a way to find a number which, when multiplied by itself, equals the given value. In other words, the square root is the inverse operation of squaring a number.

• For example: The square root of 144 is 12, because \(12 \times 12 = 144\).
• This can also be expressed as \(144^{\frac{1}{2}} = 12\).

In algebra, finding square roots of expressions allows us to simplify them more easily, as seen in our problem where we took the square root of each term within a power.

Exponents

Exponents are a shorthand way to express repeated multiplication. They follow specific rules that allow them to be manipulated in various ways for simplification purposes. When you encounter an exponent, it tells you how many times to multiply the base by itself.

• For instance, \(a^{3}\) means \(a \times a \times a\).
• This makes calculations and expressions more compact and manageable.

In algebraic simplification, you'll often need to apply rules like multiplying exponents when powers are combined. For instance, in our exercise, \((p^{2})^{\frac{1}{2}}\) simplified to \(p^{2 \times \frac{1}{2}} = p^{1}\). Understanding these principles makes working with expression simplification much easier.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form without changing the underlying values. This might involve using several algebraic rules, such as the power rule, to combine like terms, eliminate parentheses, and reduce exponents.

• To simplify, you break down expressions step by step using known mathematical principles.
• For example, starting with \(144p^{2}q^{6}\) raised to the \(\frac{1}{2}\) power, we tackled each element individually.
• First, the square root of the numerical part was found, then exponents of each variable were adjusted using the power rule.

This process results in a cleaner and simpler version of the original expression. In our exercise, the initial complex expression was reduced to \(12pq^{3}\), showcasing how simplifying helps in making expressions more digestible.