Problem 3 - Elementary and Intermediate Algebra

Question

Fill in the blanks. We read \(27^{-1 / 3}\) as " 27 to the _____ one-third power."

Step-by-Step Solution

Answer

negative

Step 1: Understanding the Problem

The given expression is a power of 27 where the exponent is negative and in fractional form, specifically \(-\frac{1}{3}\). We are asked to convert this into a verbal expression.

Step 2: Identify the Components of the Exponent

The expression \(27^{-1/3}\) has an exponent of \(-\frac{1}{3}\). We can split this exponent into a negative sign and a fraction. The fraction \(\frac{1}{3}\) indicates a 'one-third' power.

Step 3: Recognize the Negative Exponent Rule

A negative exponent, such as \(-1/3\), means that we take the reciprocal of the base raised to the positive of that exponent. Thus, \(a^{-n} = \frac{1}{a^n}\). However, the problem just requires the verbal conversion of the expression, not solving it further.

Step 4: Verbalize the Expression

When converting \(27^{-1/3}\) into words, we state the base first ('27'), mention the power ('one-third power'), and include the word for the negative ('negative'). Thus, it is read as '27 to the negative one-third power.'

Key Concepts

Fractional Exponents

Fractional exponents can seem a bit tricky at first, so let's break them down into simpler terms. When you see an exponent as a fraction, such as \(\frac{1}{3}\), it tells you to perform a specific root operation. The denominator of the fraction indicates which root to take.

For example:

• \(\sqrt{27}\) is the square root of 27, representing the \(\frac{1}{2}\) power of 27, since the denominator is 2.
• The cube root, or \(\sqrt[3]{27}\), represents the \(\frac{1}{3}\) power because the denominator is 3.

So, when you encounter \(27^{-1/3}\), the fractional \(\frac{1}{3}\) suggests a cube root. Then, the negative sign further influences the expression, which we'll delve into shortly. Understanding fractional exponents allows you to convert between radical expressions and base powers effortlessly.

Mastering this concept is foundational for tackling more advanced math topics, making complex problems simpler to handle.

Verbal Expression

Translating mathematical expressions into verbal language is essential for clear communication. When we talk about an exponent, we're describing the power to which a number, known as the base, is raised. In the expression \(27^{-1/3}\), we must effectively convey the relationship between these components.

The common format to verbalize power includes:

• The base: Always stated first, here it's '27'.
• The power: Describes the operation, such as 'one-third power' in our example due to the fractional exponent \(\frac{1}{3}\).
• Describing the negative: If the exponent is negative, we include 'negative' in the expression.

Thus, the verbal expression becomes '27 to the negative one-third power'. This format maintains consistency across mathematical discussions and written communication. Using verbal expressions accurately helps avoid confusion and promotes a more precise understanding of mathematical operations.

Reciprocal

The concept of a reciprocal often arises in mathematics when dealing with negative exponents. A reciprocal simply means flipping a number, which turns it into an inverse. In arithmetic, the reciprocal of a number \(a\) is expressed as \(1/a\).

When an exponent is negative, like in \(27^{-1/3}\), this implies taking the reciprocal of the base raised to the positive of that exponent. This follows the general rule: - \(a^{-n} = \frac{1}{a^n}\)
In our example, finding \(27^{-1/3}\) involves two steps: determining the cube root of 27 (since \(1/3\)), and then taking the reciprocal of that result because the exponent is negative. Hence, understanding the reciprocal not only helps resolve negative exponents but also simplifies calculations where this property applies.

Recognizing the power of reciprocals adds another tool to your mathematical toolbox, making various problem-solving situations more approachable.