Problem 12 - Intermediate Algebra

Question

Use radical notation to write each expression. Simplify if possible. $$ \left(16 x^{8}\right)^{1 / 2} $$

Step-by-Step Solution

Answer

\(4x^4\)

Step 1: Apply the Power of a Power Rule

To simplify the expression \((16 x^8)^{1/2}\), begin by recognizing \((a^b)^{1/c} = a^{b imes 1/c}\). Thus, \(16^{1/2} imes (x^8)^{1/2}\).

Step 2: Simplify the Square Root of 16

Next, take the square root of 16. The square root of 16 is 4, since \(4^2 = 16\). Therefore, \(16^{1/2} = 4\).

Step 3: Simplify the Radical of the Variable

For \(x^{8/2}\), you can simplify because \(8/2 = 4\). Thus, \(x^{8/2} = x^4\).

Step 4: Combine Results

Combine the results from the previous steps to get the simplified expression: \(4x^4\).

Key Concepts

Power of a Power Rule

Understanding the "Power of a Power Rule" is essential when dealing with expressions that have exponents raised to additional powers. This rule essentially tells us how to handle situations where a power is raised to another power, which can sometimes get confusing.According to the rule:

• If you have an expression like \((a^b)^{1/c}\), you can multiply the exponents: \((a)^{b \times 1/c}\).

This means you take the base number (in this case, both 16 and \(x^8\)) and raise it to the power of \(\frac{1}{2}\). This simplifies calculations by breaking down what seems complex into manageable steps.When using this rule, focus on:

• Applying the multiplication of exponents correctly.
• Remembering that each part of the expression must be dealt with separately.

In our exercise, it started with \(16^1/2 \times (x^8)^{1/2}\), simplifying it by the power of a power rule to break down the original expression systematically.

Radical Expressions

Radical expressions involve roots, such as the square root, cube root, etc. It's crucial to know these when working on expressions containing radicals.The radical sign \(\sqrt{}\) is often used to denote square roots. For example, \(\sqrt{16}\) means "square root of 16," and the result is 4 because 4 times 4 equals 16.When dealing with a simple number such as 16, the steps are:

• Find which number, when multiplied by itself, equals the original number.

For variables, the process is similar:

• For \(x^{8/2}\), simplify using division in the exponent: \(8/2 = 4\), leaving \(x^4\). This simplification helps in reducing complex expressions, making them easier to solve and understand.

Understanding radicals and their notation aids in converting expressions into simpler terms. It helps you tackle problems efficiently, using methods like those demonstrated in the example exercise.

Simplifying Exponents

Simplifying exponents is a key skill in algebra that helps to break down complex expressions into simpler, more understandable parts. This involves making use of several rules and properties, some of which have been applied in our original exercise.Several pointers are essential when simplifying exponents:

• Use the laws of exponents, such as the power of a power rule, to handle expressions efficiently.
• Remember that simplifying often involves transforming the base of the exponent to make calculations easier.

As an example, look at the exercise's simplification of \(x^{8/2}\) which resulted in \(x^4\). This is due to dividing the exponent 8 by 2, using integer division rather than complex operations.In general, always aim to reduce the exponent where possible, whether through division, multiplication, or other means, to arrive at simpler, cleaner expressions. This is how you arrive at compact results like \(4x^4\) from more complex expressions.