Problem 45 - Intermediate Algebra

Question

Evaluate each expression. See Example 3. $$ \left(-\frac{1}{216}\right)^{2 / 3} $$

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{1}{36}\).

Step 1: Simplify the Fractional Exponent

The expression \(\left(-\frac{1}{216}\right)^{2 / 3}\) means you first take the cube root and then square the result. Rewrite it as: \(\left[ \left(-\frac{1}{216}\right)^{1/3} \right]^2.\)

Step 2: Calculate the Cube Root

Find the cube root of \(-\frac{1}{216}\). The cube root of \(-216\) is \(-6\) because \((-6)^3 = -216\). Therefore, \(\left(-\frac{1}{216}\right)^{1/3} = -\frac{1}{6}\).

Step 3: Square the Result

Now, square the result from Step 2: \(\left(-\frac{1}{6}\right)^2 = \frac{1}{36}\).

Step 4: Final Result

After evaluating both the cube root and the square, the final result for the expression is \(\frac{1}{36}\).

Key Concepts

Cube Roots

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. This is the inverse operation of cubing a number. In notation, the cube root of a number \((x)\) is written as \(\sqrt[3]{x}\). For instance, if you consider the cube root of \(-216\), it's \(-6\) because \((-6) \times (-6) \times (-6) = -216\). In other words:

• The cube root of \(-216\) is \(-6\).
• The cube root function is useful when dealing with volumes or solving equations involving cubes.
• When dealing with negative numbers under a cube root, consider that an odd number of negative factors results in a negative product.

Thus, calculating cube roots helps simplify expressions where numbers are raised to fractional powers.

Squaring Numbers

Squaring a number means multiplying it by itself. This is denoted mathematically as \(x^2\), where \(x\) is any number. An important aspect is that squaring a negative number results in a positive number because a negative times a negative equals a positive. For example, when you square \(-\frac{1}{6}\), the result is \(\left(-\frac{1}{6}\right)^2 = \frac{1}{36}\). Here are some key points:

• Squaring is a common operation in geometry, particularly when dealing with areas of squares.
• The result of squaring any real number is always non-negative.
• Squaring plays a role in defining distances, as the Pythagorean theorem involves squared terms.

Overall, squaring numbers is crucial in various mathematical contexts, ensuring that transformations like geometric scaling or simplifying expressions are correctly handled.

Exponents

Exponents are a shorthand way to express repeated multiplication of the same number by itself. The general form is \(b^n\), where \(b\) is the base and \(n\) is the exponent, indicating how many times the base is used as a factor. Fractional exponents, like \(\left(-\frac{1}{216}\right)^{2 / 3}\), combine roots and powers, offering a powerful tool for simplifying complex expressions.Consider these important details:

• The exponent \(2/3\) implies taking the cube root of the base and then squaring the result.
• Fractional exponents are useful for navigating between polynomial degrees and finding roots.
• Understanding how to manipulate and simplify exponents is vital in algebra, calculus, and science.

Mastery of exponents, including fractional ones, enhances your ability to solve problems involving growth, decay, and other mathematical transformations effectively.