Problem 21 - A Graphical Approach to College Algebra

Question

Simplify each expression. Assume that all variables represent positive real numbers. $$\left(16 p^{4}\right)^{1 / 2}$$

Step-by-Step Solution

Answer

The simplified expression is \(4p^2\).

Step 1: Apply the Power of a Power Rule

When simplifying expressions with exponents, use the rule \((a^m)^n = a^{m imes n}\). Here, we have \((16p^4)^{1/2}\). Apply the exponent to both the constant and the variable separately: \(16^{1/2}\) and \((p^4)^{1/2}\).

Step 2: Simplify the Radical 16

Calculate the square root of 16. Since \(16 = 4^2\), \(16^{1/2} = 4\). This simplifies the constant part of the expression.

Step 3: Simplify the Variable Part

Apply the rule \((a^m)^n = a^{m imes n}\) to the variable exponent: \((p^4)^{1/2} = p^{4 imes 1/2}\). Simplify this to \(p^2\).

Step 4: Combine the Results

Combine the simplified results from both parts: the constant \(4\) and the variable \(p^2\). Thus, the expression \((16p^4)^{1/2}\) simplifies to \(4p^2\).

Key Concepts

Power of a Power Rule

When dealing with expressions that involve exponents raised to another power, the Power of a Power Rule is your go-to tool. This rule states that \[(a^m)^n = a^{m \times n}.\]Let's break this down: Imagine you have a number or a variable raised to an exponent, and then that whole thing is raised to another exponent. Instead of calculating each power separately, you can simply multiply the exponents together.

• The base (in our case, the number or the variable) remains the same.
• The exponents are multiplied to consolidate the expression.

For example, if you have \((x^3)^2\), you can simplify this by multiplying the exponents: \(x^{3 \times 2} = x^6\).
It makes computations much easier by reducing the steps needed when simplifying complex expressions.
In our exercise, \((p^4)^{1/2}\) becomes \(p^{4 \times 1/2} = p^2\), demonstrating this useful rule.

Square Root

The square root is a common and essential mathematical operation. It is fundamentally the opposite, or inverse, of squaring a number.
To find the square root of a number, you look for a value that, when multiplied by itself, equals the original number. Say you're working with 16. The square root, noted with the symbol \(\sqrt{}\), is simply 4 because \(4 \times 4 = 16\). For powers, the square root can be expressed with an exponent of \(1/2\).
Remember:

• Finding the square root of a number simplifies the expression.
• When dealing with radicals, express square roots in their simplest form.

In our case, \(16^{1/2} = 4\). This step ensures that the numerical component of the expression is as simple as possible, making the whole expression easier to work with and understand.

Exponents

Exponents are powerful tools in mathematics that indicate repeated multiplication. They allow us to compress long multiplication sequences into more manageable expressions. For example, instead of writing \(p \times p \times p \times p\), we can simply write \(p^4\). The number 4 here is the exponent, indicating that the base (\(p\)) is multiplied by itself four times.
Understanding some key properties of exponents can simplify complex mathematical expressions:

• Product of Powers: For the same base, add the exponents \( (a^m \times a^n = a^{m+n}) \).
• Quotient of Powers: Subtract the exponents \( (a^m / a^n = a^{m-n}) \).
• Power of a Power: We multiply the exponents \( (a^m)^n = a^{m \times n} \).

In our exercise, we used the power of a power property to simplify our expression. Specifically, \((p^4)^{1/2}\) becomes \(p^2\). Applying these principles helps streamline mathematical calculations, making exponents essential in simplifying various expressions.