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Chapter 5: Problem 57

Simplify each expression. Write answers using positive exponents.. $$ -h^{-1} $$

Short Answer

Expert verified
-\frac{1}{h}

Step by step solution

01

Recognize the Inverse Exponent

The expression given is \(-h^{-1}\). The inverse exponent indicates a reciprocal, meaning the term can be rewritten in another form.
02

Apply the Rule of Negative Exponents

The rule of negative exponents states: \(a^{-n} = \frac{1}{a^n}\). Thus, \(-h^{-1}\) becomes \(-\frac{1}{h^1}\).
03

Express in Simplified Form

Since raising to the power of 1 does not change the base, \(h^1\) is equivalent to \(h\). Therefore, the expression simplifies to \(-\frac{1}{h}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
Simplifying algebraic expressions is like cleaning up a messy room. We take what's given, apply certain rules, and make it look more organized. In our example, we're simplifying \(-h^{-1}\). The goal here is to transform this expression to make it clearer and use only positive exponents, which is typically more standard and preferable in mathematical writing. Breakdown processes with negative exponents help to ensure that mathematical expressions remain straightforward and easily understandable.
  • Identify components within the expression that can be rewritten.
  • Use rules and mathematical operations to transform the expression efficiently.
  • Double-check to ensure the final expression is simplified as much as possible.
Thus, through simplifying, we aim to convey the same information using the simplest form possible.
Positive Exponents
In the realm of exponents, positive exponents are your friendly neighbors. They make expressions straightforward and easy to manage. Positive exponents indicate how many times the base number is multiplied by itself. For example, \(h^1\) simply means the base \(h\) is used once, which doesn't alter the base. This rule of using positive exponents is crucial for maintaining simplicity in expressions and ensuring clarity in calculations.
  • Positive exponents help avoid confusion when solving equations.
  • The standard form is generally presented with positive exponents.
  • They offer a clean and traditional structure for mathematical expressions.
By ensuring that all exponents we use are positive, we keep our math training wheels on and make problems easier to tackle.
Reciprocals
Reciprocals are fascinating aspects of numbers and play an essential role in math. Simply put, the reciprocal of a number is what you get when you flip the number upside down. For example, the reciprocal of \(h\) is \(\frac{1}{h}\). In our exercise, the reciprocal came into play when dealing with the negative exponent. A negative exponent hints that you need to find the reciprocal of the base.
The idea of reciprocals is quite helpful when doing division or simplifying expressions:
  • It helps convert negative exponents into positive ones by flipping the base.
  • When you multiply a number by its reciprocal, the result is always 1.
Understanding reciprocals can make many algebraic manipulations more intuitive and are especially useful in solving equations and simplifying expressions like in our example.

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