Problem 57 - Applied Calculus

Question

\(57-70 .\) Write each expression in power form \(a x^{b}\) for numbers \(a\) and \(b\). $$ \frac{4}{x^{5}} $$

Step-by-Step Solution

Answer

The expression \( \frac{4}{x^5} \) is written in power form as \( 4x^{-5} \).

Step 1: Identify the expression format

The given expression is a fraction: \( \frac{4}{x^5} \). Here, \(4\) is in the numerator, and \(x^5\) is in the denominator.

Step 2: Rewrite the denominator with negative exponent

From the rules of exponents, \( x^{-b} = \frac{1}{x^b} \). Using this rule, \( \frac{1}{x^5} \) can be rewritten as \( x^{-5} \).

Step 3: Express the entire fraction in power form

Using the result from Step 2, \( \frac{4}{x^5} \) can be rewritten by applying \( 4 \times x^{-5} \). This converts the expression to power form: \( 4x^{-5} \).

Key Concepts

Fraction to Negative Exponent

When working with exponents and algebraic expressions, it’s often useful to convert fractions into forms involving negative exponents. This conversion makes computations and further algebraic manipulations simpler.
Let's consider the fraction \( \frac{4}{x^5} \). In this expression, the base \( x \) is raised to the power of 5 in the denominator.
Here's how it works:

• The power rule for exponents states that a term in the denominator with a positive exponent can be shifted to the numerator by negating its exponent.
• This means \( \frac{1}{x^5} = x^{-5} \). So, the fraction \( \frac{4}{x^5} \) would then become \( 4x^{-5} \). This step of converting the fraction to negative exponents can simplify algebraic expressions, especially when multiple terms need to be combined or compared.

By adopting this approach, you gain flexibility in solving equations and simplifying expressions, making the math much more manageable.

Exponent Rules

Exponent rules are foundational in simplifying expressions and solving equations in algebra. These rules govern how to handle powers of numbers and variables effectively.
Here are some of the key exponent rules that come in handy:

• Product of Powers: When multiplying like bases, you add the exponents, \( x^a \times x^b = x^{a+b} \).
• Quotient of Powers: When dividing like bases, you subtract the exponents, \( \frac{x^a}{x^b} = x^{a-b} \).
• Power of a Power: When raising a power to another power, you multiply the exponents, \( (x^a)^b = x^{a \cdot b} \).
• Zero Exponent: Any number (except zero) raised to the power of zero equals one, \( x^0 = 1 \).
• Negative Exponents: A negative exponent indicates a reciprocal, meaning \( x^{-a} = \frac{1}{x^a} \).

These rules not only help in simplifying expressions like turning \( \frac{4}{x^5} \) into \( 4x^{-5} \), but also enable solving complex algebraic problems by breaking them into simpler terms.

Algebraic Expressions

Algebraic expressions form the backbone of algebra and are a combination of variables, numbers, and operations. Understanding these expressions is crucial for solving equations, modeling real-life situations, and performing calculations.
An algebraic expression may include:

• Constants: Numbers that stand alone without variables, e.g., \( 4 \) in our expression.
• Variables: Symbols representing unknown values, such as \( x \).
• Exponents: Indicate how many times a number is multiplied by itself, like the \( x^{-5} \) portion.

The expression \( 4x^{-5} \) is called monomial because it consists of a single term. These expressions can be used in various operations, such as:

• Addition and Subtraction: Combining like terms.
• Multiplication and Division: Using exponent rules to simplify.
• Evaluating: Substituting values for variables.

Mastering algebraic expressions equips you with the tools to tackle more complex problems and gives you insight into the structure of mathematical arguments and solutions.