Problem 134 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(x^{4}-16\) is factored completely as \(\left(x^{2}+4\right)\left(x^{2}-4\right)\)

Step-by-Step Solution

Answer

The given statement \(8^{-\frac{1}{3}}=-2\) is false. The true statement is \(8^{-\frac{1}{3}} = \frac{1}{2}\).

Step 1: Calculating Exponent

First calculate the value of \(8^{-\frac{1}{3}}\). A negative exponent means to divide 1 by the base raised to the absolute value of the exponent. A fractional exponent \(a^(m/n)\) is equivalent to the nth root of a raised to the m. Therefore, this can simplify to \(\frac{1}{\sqrt[3]{8}}\).

Step 2: Determining the Value

Compute the cube root of 8, which is 2. Therefore, the expression simplifies to \( \frac{1}{2} \)

Step 3: Comparing with Given Value

Compare the simplified value with -2. The value \( \frac{1}{2} \) is not equal to -2. Thus, the given statement is false.

Step 4: Correct the Statement

To make the statement true, replace -2 with \( \frac{1}{2} \) in the original statement, which results in the true statement \(8^{-\frac{1}{3}} = \frac{1}{2}\).

Key Concepts

Fractional Exponents

Fractional exponents are forms of exponents where the power is a fraction, instead of a whole number. These are incredibly useful and occur frequently when dealing with roots and powers.
For instance, the expression \(a^{m/n}\) can be interpreted as the "nth root of \(a\), raised to the 'm' power". This brings about flexibility in math by allowing roots and powers to be expressed in a uniform manner.
In the example \(8^{-rac{1}{3}}\), the fraction \(-\frac{1}{3}\) indicates two operations: a cube root and a reciprocal, due to the negative sign. Understanding this helps greatly in converting between radicals and exponents.

Cube Roots

Cube roots are a specific type of root, akin to square roots, but pertaining to cubes. A cube root of a number \(a\) is another number \(b\) such that \(b^3 = a\). A simple way to visualize this is to imagine a perfect cube.
The cube root is the measure of the edge of this cube. In our example, finding the cube root of 8 simplifies to finding which number, when multiplied by itself three times, results in 8. Thus, \(\sqrt[3]{8}\) equals 2, because \(2^3 = 8\).
Recognizing the cube root allows for simplification of complex expressions, especially those involving fractional exponents.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to work with or compare. This process involves operations like performing arithmetic, reducing fractions, and correctly applying mathematical principles.
In the exercise, simplifying \(8^{-rac{1}{3}}\) is about understanding and translating the negative fractional exponent to easy-to-understand forms (like roots and reciprocals). This turns into calculating \(\frac{1}{\sqrt[3]{8}}\) which leads us to \(\frac{1}{2}\), a much simpler expression.
Simplifying requires recognizing potential simplifications, applying mathematics correctly, and always reevaluating to ensure the simplest possible form is achieved.

True or False Statements

Mathematics often involves evaluating statements to determine if they are true or false. Understanding when a statement is false and knowing how to correct it is crucial in problem-solving.
In our example, the statement \(8^{-rac{1}{3}} = -2\) was false. We corrected it by accurately simplifying the left-hand side and comparing it to -2. When it didn't match, we provided the correct value, \(\frac{1}{2}\).
Evaluating true or false statements can strengthen understanding of concepts and improve verification skills, ensuring students can validate their solutions accurately. It helps to not only identify errors but also understand the corrections needed for the true outcome.