Problem 60 - Brief Applied Calculus

Question

Write each expression in power form \(a x^{b}\) for numbers \(a\) and \(b\). $$ \frac{6}{\sqrt{4 x^{3}}} $$

Step-by-Step Solution

Answer

The expression is written as \(3x^{-3/2}\).

Step 1: Simplify the Square Root

The expression inside the square root is \(4x^3\). We can use the property \(\sqrt{a} = a^{1/2}\) to represent the square root, thus \(\sqrt{4x^3} = (4x^3)^{1/2}\).

Step 2: Separate the Base Terms

We can apply the power of a product rule, which states \((ab)^c = a^c b^c\). Thus, split \((4x^3)^{1/2}\) into \(4^{1/2} \times (x^3)^{1/2}\).

Step 3: Simplify Each Term

Calculate \(4^{1/2} = 2\) and simplify \((x^3)^{1/2} = x^{3/2}\). This gives us \(2x^{3/2}\).

Step 4: Write the Initial Expression Using the Simplified Denominator

We now have \(\frac{6}{2x^{3/2}}\). Simplify this expression by dividing both the numerator and the constant part of the denominator by 2, which gives \(\frac{3}{x^{3/2}}\).

Step 5: Express Using Negative Exponents

Use the property \(\frac{1}{x^{a}} = x^{-a}\) to rewrite the expression as \(3x^{-3/2}\). This is now in the form \(ax^b\), where \(a = 3\) and \(b = -3/2\).

Key Concepts

Rational Exponents

Rational exponents are a way to express powers and roots together. By understanding this concept, calculations involving roots and powers become simpler and often more intuitive. Imagine you have the square root of a number, say 4. You can write this as a rational exponent by expressing it as a power: \(4^{1/2}\). Here, "\(1/2\)" is a rational exponent representing a square root.

This approach extends beyond square roots. For example, a cube root can be represented with an exponent of \(1/3\). When dealing with expressions like \((4x^3)^{1/2}\), each component within the parentheses takes the \(1/2\) exponent. This use of rational exponents is vital for efficient manipulation of expressions.

Power of a Product Rule

The power of a product rule is a handy tool when simplifying expressions involving exponents. This rule states \((ab)^c = a^c \cdot b^c\). It enables the separation of terms under a common power, allowing us to simplify each term individually.

In our example for \((4x^3)^{1/2}\), we utilize this rule to split the expression into two parts: \(4^{1/2}\) and \((x^3)^{1/2}\).

• For \(4^{1/2}\), it simplifies directly to 2, given that the square root of 4 is 2.
• For \((x^3)^{1/2}\), applying the rule results in \(x^{3/2}\).

Using this rule is especially useful when handling multiple variables and coefficients together in an exponential expression.

Negative Exponents

Understanding negative exponents is crucial for rewriting and simplifying expressions. A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent.

For instance, \(x^{-a}\) is equivalent to \(\frac{1}{x^a}\). This conversion is quite useful when an expression needs to be rewritten or rearranged, like in our exercise. In order to express \(\frac{3}{x^{3/2}}\) in a power form, we convert \(1/x^{3/2}\) into \(x^{-3/2}\). This consistent method of handling exponents ensures that expressions remain in a simplified, uniform format.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves reducing an expression to its most basic and manageable form. The goal is to minimize the complexity while maintaining the expression's value.

In this context, simplifying often includes combining similar terms, reducing fractions, and converting exponents. In the provided solution, simplifying steps include reducing \(\frac{6}{2x^{3/2}}\) to \(\frac{3}{x^{3/2}}\) by dividing the numerator and denominator by 2.

• This simplification helps in achieving the final form \(3x^{-3/2}\), making it easier to interpret and work with.
• By expressing the denominator as a negative exponent, we neatly align the expression into a single power form.

Grasping how to simplify accurately can expose the underlying structure of an expression, making further mathematical operations straightforward.