Problem 65 - Applied Calculus

Question

Write each expression in power form \(a x^{b}\) for numbers \(a\) and \(b\). $$ \frac{5 x^{2}}{\sqrt{x}} $$

Step-by-Step Solution

Answer

The expression is equal to \(5 x^{\frac{3}{2}}\).

Step 1: Identify the Expression in Power Form

Begin by identifying how you can express the elements of the given expression in power form. The expression is \(\frac{5 x^{2}}{\sqrt{x}}\). You can rewrite \(\sqrt{x}\) as \(x^{\frac{1}{2}}\) because the square root of any number is equivalent to raising it to the power of \(\frac{1}{2}\).

Step 2: Rewrite the Expression with Common Base

Substitute \(\sqrt{x}\) with \(x^{\frac{1}{2}}\) in the expression: \(\frac{5 x^{2}}{x^{\frac{1}{2}}}\). Now both terms in the fraction have the same base, \(x\).

Step 3: Apply the Quotient Rule of Exponents

The quotient rule of exponents states that \(\frac{x^{m}}{x^{n}} = x^{m-n}\). Apply this to the expression \(\frac{5 x^{2}}{x^{\frac{1}{2}}}\), where \(m = 2\) and \(n = \frac{1}{2}\). This gives you \(x^{2 - \frac{1}{2}}\), which simplifies to \(x^{\frac{3}{2}}\).

Step 4: Express in the Form \(a x^{b}\)

Now that you have simplified the expression using the rules of exponents, combine the constant with the simplified term: \(5 \times x^{\frac{3}{2}}\). Thus, the expression in the form \(a x^{b}\) is \(5 x^{\frac{3}{2}}\).

Key Concepts

Power Form

Expressions in mathematics are often simplified into a more universally understandable form. One such form is the power form. Writing an expression in power form means expressing it as \( a x^{b} \), where \( a \) is a coefficient and \( b \) represents the power to which \( x \) is raised. This form allows for easier manipulation and comparison of expressions. For example, the expression \( \sqrt{x} \) can be written in power form as \( x^{\frac{1}{2}} \). This conversion makes further calculations simpler, especially when working with products or quotients involving similar bases. Transforming to power form is an essential skill because it sets the groundwork for applying exponent rules more efficiently.

Consistently utilizing power form helps maintain clarity and uniformity across mathematical expressions, making it easier to apply algebraic operations such as multiplication, division, and simplification.

Quotient Rule

The quotient rule is a fundamental exponent rule that simplifies expressions where one power is divided by another, provided they have the same base. The rule states that \( \frac{x^{m}}{x^{n}} = x^{m-n} \). This property allows for the subtraction of the exponents when dividing two like bases.

Applying the quotient rule helps in reducing the complexity of expressions. For instance, when faced with the expression \( \frac{5 x^{2}}{x^{\frac{1}{2}}} \), both terms have the base \( x \). Using the quotient rule, the exponents are subtracted: \( 2 - \frac{1}{2} = \frac{3}{2} \). Thus, it simplifies to \( x^{\frac{3}{2}} \). This simplification not only neatly condenses the expression but also prepares it for further arithmetic or algebraic operations.

Understanding and applying the quotient rule is a significant step in expression manipulation, resulting in cleaner and more manageable terms.

Expression Simplification

Simplifying a mathematical expression involves reducing it to its most concise and manageable form without changing its value. This is achieved through various rules and operations such as combining like terms, managing constants, and applying exponent rules.

In the original exercise, the simplification process begins once the expression is rewritten to a uniform power form. After transforming \( \sqrt{x} \) into \( x^{\frac{1}{2}} \), and applying the quotient rule, we find a new expression \( 5 x^{\frac{3}{2}} \). Simplification like this makes it easier to interpret and solve problems that include the expression, whether in further mathematical contexts or real-world applications.

Simplification plays a crucial role in mathematics by enabling the handling of complex calculations in a more straightforward manner. It aids in revealing the essential characteristics of expressions and can make solving equations or inequalities more direct and less prone to error.