Problem 37 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(-3^{-3}\)

Step-by-Step Solution

Answer

The simplified form of \(-3^{-3}\) is \(\frac{-1}{27}\).

Step 1: Understanding Negative Exponents

When you have an expression with a negative exponent, like \[-a^{-n}\]you can rewrite it as a fraction:\[\frac{-1}{a^n}\] This means you are taking the reciprocal of the base and changing the sign of the exponent. Apply this rule to simplify the given expression.

Step 2: Apply Negative Exponent Rule to \(-3^{-3}\)

Rewrite \(-3^{-3}\) using the rule from Step 1:\[-3^{-3} = \frac{-1}{3^3}\]The negative sign remains in the numerator, and the base 3 is raised to the positive power in the denominator.

Step 3: Evaluate the Positive Exponent

Now, evaluate \(3^3\).This involves multiplying 3 by itself two times:\[3 \times 3 \times 3 = 27\]Substitute this back into the fraction we obtained in Step 2.

Step 4: Combine to Get the Final Answer

Substitute 27 into the solution from Step 2 to get the final simplified answer:\[\frac{-1}{27}\]This is the simplified form, and it does not contain any negative exponents.

Key Concepts

Simplifying Expressions

When simplifying expressions, the goal is to rewrite them in their simplest form. This often means removing unnecessary components or transforming the expression to a more basic version. In the context of exponents, especially negative exponents, it requires a deep understanding of what each part of the expression represents.

• Identifying the components of an expression is the first step. For example, in \(-3^{-3}\), the negative exponent indicates the need for simplification.
• Using mathematical properties, like the reciprocal of a base, helps in the simplification process. Simply put, simplifying an expression allows easier computation and interpretation.
• By eliminating negative exponents, we convert the expression into a more standardized form, making further operations straightforward.

Remember, a well-simplified expression enables both efficiency in calculations and clarity in comprehension.

Reciprocal of a Base

The reciprocal of a base is a fundamental concept in handling negative exponents. Reciprocals flip the numerator and denominator, transforming \(a^{-n}\) into \(\frac{1}{a^n}\).

• When dealing with terms like \(-3^{-3}\), the reciprocal helps in understanding how we can avoid negative exponents. The base \(3\) becomes the denominator, simplified as \(3^3\) without the negative sign.
• This method flips the perspective, allowing us to manipulate and express the same quantity under positive exponents, which are more intuitive.
• After finding the reciprocal, simplify further by evaluating the positive exponents. For \(3^3\), this means computing \(3 \times 3 \times 3\), resulting in \(27\).

Using reciprocals not only simplifies the expression but conforms to the mathematics convention of avoiding negative exponents.

Fraction Evaluation

Evaluating a fraction involves performing the mathematical calculations needed to express the fraction in its simplest form. After rewriting \(-3^{-3}\) as \(\frac{-1}{27}\), the next step is fraction evaluation.

• This process means substituting the calculated value of the base with its respective power, converting the expression into a clear fraction without negative exponents.
• Calculating \(3^3\) to get \(27\) makes the fraction easier to handle as every component is a positive integer.
• By ensuring both the numerator and the denominator are simple, as \(-1\) and \(27\), respectively, it makes further arithmetic operations, if needed, more direct.

Understanding how to evaluate fractions properly prepares students to tackle more complex expressions with confidence and accuracy.