Problem 118 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ \left(16 a^{4}\right)^{-1 / 2} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{1}{4a^2} \).

Step 1: Apply the Negative Exponent Rule

The negative exponent rule states that \( a^{-n} = \frac{1}{a^n} \). We can apply this to the expression \( \left(16a^4\right)^{-1/2} \), which gives us \( \frac{1}{\left(16a^4\right)^{1/2}} \).

Step 2: Apply the Power of a Product Rule

The power of a product rule states that \((ab)^n = a^n b^n\). Therefore, we rewrite the expression as you simplify the terms inside: \( \left(16a^4\right)^{1/2} = 16^{1/2} \cdot (a^4)^{1/2} \).

Step 3: Simplify the Square Roots

First, find \( 16^{1/2} \), which is 4, because \(4^2 = 16\). Then simplify \( (a^4)^{1/2} \) to \( a^{4/2} = a^2 \). With these calculations, \( \left(16a^4\right)^{1/2} \) simplifies to \( 4a^2 \).

Step 4: Simplify the Entire Fraction

Substitute the simplified terms \(4a^2\) back into the expression: \( \frac{1}{4a^2} \). Therefore, the simplified form of the expression \( \left(16a^{4}\right)^{-1 / 2} \) is \( \frac{1}{4a^2} \).

Key Concepts

Power of a Product Rule

When simplifying expressions, understanding the Power of a Product Rule is very helpful. This rule tells us how to deal with exponents when they are applied to a product of two or more terms. According to this rule, \((ab)^n = a^n \cdot b^n\). This means that the exponent \(n\) is applied to each factor within the product individually.

For example, if we have a product \((16a^4)\) raised to the power of \(\frac{1}{2}\), we can split it into two parts. We first apply the power to the numeric part, which is \(16\), and then to the variable part, which is \(a^4\). Hence, we transform \( (16a^4)^{1/2} \) into \( 16^{1/2} \cdot (a^4)^{1/2} \).

This simplification can make complex expressions much easier to handle, especially when combined with other algebraic rules.

Square Roots simplification

Simplifying square roots is a key skill in working with algebraic expressions. In our example, the expression \(16^{1/2}\) is the square root of 16. Square roots question what number multiplied by itself gives us the number under the root. For \(16\), the answer is \(4\), because \(4\times 4 = 16\).

When dealing with variables, such as \((a^4)^{1/2}\), we follow a similar logic. Here, the square root of \(a^4\) simplifies to \(a^{4/2}\), or \(a^2\), because the exponent is divided by \(2\). It's important to understand how to handle both numerical and variable components within expressions to simplify problems accurately.

This tactic of breaking down numbers and variables alike by their square roots ensures that you're applying consistent rules across the board.

Algebraic expression simplification

Simplifying algebraic expressions is often necessary to make them easier to work with or to find solutions. In the given exercise, understanding several rules happens to be crucial in arriving at a simpler form.

At first, we used the Negative Exponent Rule to flip the fraction, moving from \((x)^{-1/2}\) form to \(\frac{1}{x^{1/2}}\). Then, we applied the Power of a Product Rule, which allowed us to separately address each component within the parenthesis. Also, using square root simplifications helped break variables into simpler, more understandable pieces like \(4a^2\).

Finally, integrating these separate simplifications provided the streamlined result: \(\frac{1}{4a^2}\). By applying each rule strategically, complex expressions become less intimidating, guiding you to intelligible outcomes.