Problem 63 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(\frac{h^{-5}}{h^{2}}\)

Step-by-Step Solution

Answer

\( \frac{1}{h^{7}} \)

Step 1: Apply the Division Rule for Exponents

According to the division rule for exponents, when dividing like bases, you subtract the exponent in the denominator from the exponent in the numerator. In this case, the expression is \( \frac{h^{-5}}{h^{2}} \). The subtracting of exponents becomes \( h^{-5 - 2} = h^{-7} \).

Step 2: Convert Negative Exponent to Positive

In order to express the result with positive exponents, recall the rule that \( a^{-n} = \frac{1}{a^{n}} \). Applying this to \( h^{-7} \) gives \( \frac{1}{h^{7}} \).

Key Concepts

Exponent Rules

Exponent rules are fundamental in simplifying algebraic expressions involving powers. These rules help in performing arithmetic operations on expressions with exponents, facilitating the simplification process. One of the key rules to remember is the division rule for exponents, which states:

• When you divide two expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator.
• For example, in the expression \( \frac{h^{-5}}{h^2} \), the exponents are \(-5\) in the numerator and \(2\) in the denominator. Using the division rule, you subtract the exponents: \(-5 - 2 = -7\), resulting in \(h^{-7}\).

Understanding these rules helps ensure accuracy and efficiency when working with complex expressions and different bases, ensuring that results are always the simplest form possible.

Negative Exponents

Working with negative exponents can be tricky at first, but once you grasp the concept, they become much easier to handle. A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent:

• The rule for transforming a negative exponent into a positive one is given by: \( a^{-n} = \frac{1}{a^{n}} \). This means that any negative exponent can be converted to a positive exponent by taking the reciprocal.
• For our earlier example, \(h^{-7}\) can be rewritten as \(\frac{1}{h^7}\), converting the negative exponent to a positive one.
• This transformation is valuable because it generally provides a simpler or more interpretable form, especially when performing further calculations or applying other mathematical rules.

So, whenever you encounter a negative exponent, just remember that it denotes division — specifically, dividing 1 by the base raised to the positive version of that exponent.

Simplifying Expressions

Simplifying expressions is all about taking a complex mathematical expression and rewriting it in a simpler or more comprehensible form. This often involves using various algebraic rules, such as exponent rules, to cut through the complexity.

• Simplifying can involve combining like terms, cancelling terms, and applying arithmetic rules such as addition, subtraction, multiplication, and division.
• In algebraic expressions with exponents, simplifying enables the cleaning up of terms so that results are easy to interpret or further manipulate mathematically.
• With the expression \( \frac{h^{-5}}{h^2} \), we've used the division rule for exponents followed by the rule for negative exponents to simplify it to \(\frac{1}{h^7}\).

Effective simplification results in expressions that are not only easier to read but also easier to solve, making further algebraic manipulations more straightforward.