Problem 3 - Intermediate Algebra

Question

We read \(27^{-1 / 3}\) as " 27 to the _______one-third power.”

Step-by-Step Solution

Answer

27 to the negative one-third power.

Step 1: Understanding the Expression

The exercise asks us to interpret the expression \( 27^{-1/3} \). The notation \( 27^{-1/3} \) indicates that we are taking the number 27 to a particular fractional power.

Step 2: Identifying Fractional Power

In the expression \( 27^{-1/3} \), the fraction \(-1/3\) is the power to which the number 27 is raised. A fractional power like \( \frac{1}{3} \) typically refers to a root, specifically the cube root in this case.

Step 3: Interpreting the Fractional Exponent

A negative exponent indicates the reciprocal of the base raised to the corresponding positive power. Specifically, \( a^{-n} = \frac{1}{a^n} \). Therefore, \( 27^{-1/3} = \frac{1}{27^{1/3}} \).

Step 4: Converting to Roots

The expression \( 27^{1/3} \) means the cube root of 27. Thus, \( 27^{-1/3} \) is the reciprocal of the cube root of 27.

Step 5: Filling in the Blank

The phrase "27 to the _______ one-third power" is completed by identifying that \( 27^{-1/3} \) reads as "27 to the negative one-third power." Thus, the blank should be filled with "negative."

Key Concepts

Negative Exponents

Negative exponents might seem intimidating at first, but they are quite simple once you understand their meaning. The negative sign indicates that instead of multiplying, you are actually dividing by the base raised to the positive of that power. In other words, for any base, a negative exponent tells you to take the reciprocal of the base raised to the corresponding positive exponent.

For example, if you have an expression such as \(a^{-n}\), it can be rewritten as:

• \(a^{-n} = \frac{1}{a^n}\)

This means that instead of multiplying \(a\) by itself \(n\) times, you are dividing 1 by \(a\) multiplied by itself \(n\) times. For instance, \(10^{-2} = \frac{1}{10^2} = \frac{1}{100}\).

This concept not only simplifies complex expressions but also helps when working with fractional exponents, where the base involves roots.

Cube Roots

Cube roots are another key concept, especially when dealing with fractional exponents. The cube root of a number is a value that, when multiplied by itself twice more (or cubed), gives the original number. It is denoted as \(a^{1/3}\), where \(a\) is your original number.

For example, the cube root of 27 is 3 because \(3 \times 3 \times 3 = 27\). This means that \(27^{1/3} = 3\).

Understanding cube roots is essential when interpreting an expression like \(27^{-1/3}\). Here, you first calculate the cube root (getting 3), and then take its reciprocal due to the negative exponent.

When you become comfortable with the concept of cube roots, you'll find them helpful across various mathematical fields, such as solving equations and simplifying radical expressions. Being familiar with cube roots also aids in recognizing patterns among similar exponential expressions.

Reciprocal

The reciprocal of a number is another term for its multiplicative inverse. It involves flipping the numerator and the denominator. Essentially, if you have a number \(a\), its reciprocal is \(\frac{1}{a}\).

Reciprocals are particularly significant when working with negative exponents. As mentioned earlier, a negative exponent, such as \(-1/3\), indicates the reciprocal of the base raised to the positive version of that exponent.

For instance, in the expression \(27^{-1/3}\), the negative exponent tells us to take the reciprocal of the cube root of 27, which is 3. Therefore, \(27^{-1/3} = \frac{1}{3}\).

Reciprocals are a handy tool not just in simplifying expressions with exponents, but also in solving fractions and equations, where you need to invert values for computation. Understanding reciprocals will enhance your ability to manipulate and simplify complex mathematical expressions.