Problem 69 - Applied Calculus

Question

Write each expression in power form \(a x^{b}\) for numbers \(a\) and \(b\). $$ \frac{\sqrt{36 x}}{2 x} $$

Step-by-Step Solution

Answer

The expression is \(3x^{-1/2}\).

Step 1: Simplify the Square Root

Begin by simplifying the square root in the numerator. The expression under the square root is \(36x\), which can be written as \(\sqrt{36} \times \sqrt{x}\) using the property that \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\). This simplifies to \(6\sqrt{x}\) because \(\sqrt{36} = 6\).

Step 2: Write \(\sqrt{x}\) in Exponent Form

The square root of \(x\) can be expressed in exponent form as \(x^{1/2}\). Therefore, \(6\sqrt{x}\) becomes \(6x^{1/2}\).

Step 3: Formulate the Entire Fraction

Now, the entire expression can be rewritten as \(\frac{6x^{1/2}}{2x}\). Simplify the fraction by separating the coefficients and the variables: \(\frac{6}{2} \cdot \frac{x^{1/2}}{x}\). This simplifies to \(3\cdot x^{-1/2}\) by dividing the coefficients and applying the rule of exponents that says \(x^{a}/x^{b} = x^{a-b}\).

Step 4: Identify \(a\) and \(b\)

The simplified expression in power form is \(3x^{-1/2}\). Here, \(a = 3\) and \(b = -1/2\).

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in algebra that helps make complex problems easier to solve. It involves reducing expressions to their simplest form without changing their value. In the context of the given problem, the expression \( \frac{\sqrt{36x}}{2x} \) is simplified. The first step is to simplify the numerator by breaking down the square root, separating factors to make them easy to manage.
For example, \( \sqrt{36x} \) can be decomposed into \( \sqrt{36} \times \sqrt{x} \). This property works because the square root of a product is the product of the square roots.

The process involves:

• Breaking down grouped factors within the square root.
• Reducing components to their simplest form.

This simplification process is crucial as it makes the handling of expressions more straightforward and sets the stage for further operations such as expressing components in exponent form, which is the next step.

Square Roots

Square roots are an intriguing concept in mathematics with roots in simplifying expressions and using formulas. They let you find the number that, when multiplied by itself, results in the original number. In our exercise, the square root \( \sqrt{36x} \) is separated to streamline calculations. The square root of 36 (a perfect square) is 6, and that makes it easy to simplify. This results in \( 6\sqrt{x} \) initially.

Key things to remember:

• Factor the number under the square root for easier simplification.
• If there's a perfect square, resolve it so that you can reduce the overall expression.

Square roots are a versatile part of algebraic expressions, engaging in tasks from simplifying expressions to solving equations. Simplifying them effectively is a useful skill that benefits from frequent practice.

Laws of Exponents

The laws of exponents are fundamental rules that help in manipulating powers of numbers efficiently. They are especially handy when working with algebraic expressions and easing complex calculations involving powers. In our exercise, once the square root is simplified to \( 6\sqrt{x} \), it is expressed in exponent form as \( 6x^{1/2} \). This step simplifies later divisions by allowing the application of exponents rules effectively.

These rules can include:

• \( x^a \times x^b = x^{a+b} \)
• \( \frac{x^a}{x^b} = x^{a-b} \)
• \((x^a)^b = x^{a\cdot b} \)

In our context, \( \frac{6x^{1/2}}{2x} \) utilizes the exponent rules to reach \( 3x^{-1/2} \). Understanding these rules allows simplification into a manageable form, where \( a = 3 \) and \( b = -1/2 \), aligned with standard power form, further easing any subsequent algebraic manipulation.