Problem 40 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ (-27)^{-1 / 3} $$

Step-by-Step Solution

Answer

The expression (-27)^{-1/3} evaluates to -1/3.

Step 1: Understand the Negative Exponent

The expression (-27)^{-1/3} involves a negative exponent. A negative exponent means taking the reciprocal of the base raised to the positive of that exponent. In this case, the expression (-27)^{-1/3} can be rewritten as \( \frac{1}{(-27)^{1/3}} \).

Step 2: Evaluate the Cubic Root

The positive exponent 1/3 signifies a cube root. You need to find the cube root of -27, which is the number that, when multiplied by itself three times, equals -27. The cube root of -27 is -3. This is because (-3) imes (-3) imes (-3) = -27.

Step 3: Calculate the Final Expression

Substitute the cube root back into the expression. You now have \( \frac{1}{(-3)} \). Simplifying this expression gives \( -\frac{1}{3} \).

Key Concepts

Cube Root

Understanding the concept of a cube root is crucial to solving many mathematical problems. A cube root is the inverse operation of cubing a number.
In simpler terms, if you know the cube of a number, finding its cube root will yield back the original number.
For example, if you have a number like -27 and you want to find its cube root, you're looking for a number that when multiplied by itself three times, will give you -27.
To calculate this, consider:

• You want a number that, when multiplied to itself twice more, results in -27.
• In this case, that magic number is -3, because:ewline\((-3) \times (-3) \times (-3) = -27\)

Notice that the cube root of a negative number is also negative, as three negatives multiply to yield a negative result. The idea of cube roots extends from basic numbers to algebraic expressions and is fundamentally important when solving various equation types.

Reciprocal Function

Reciprocal functions are a fascinating aspect of mathematics that often come into play when working with exponents. When you have a number raised to a negative exponent, this tells you to take the reciprocal of that number's positive exponent.
The notion of reciprocals is easy: the reciprocal of a number is 1 divided by that number. For example:

• The reciprocal of 2 is \( \frac{1}{2} \).
• Similarly, the reciprocal of -3 is \( \frac{1}{-3} \) or \(-\frac{1}{3} \).

In the context of the given expression \((-27)^{-1/3}\), the negative exponent \(-1/3\) suggests that instead of directly taking the cube root of -27, you first need its reciprocal: \( \frac{1}{(-27)^{1/3}} \). This rule transforms a complex-looking expression into something more manageable.
Understanding reciprocal functions will perform wonders in simplifying and solving various mathematical expressions.

Simplifying Expressions

Simplifying expressions means making them as straightforward as possible without changing their value. This process is key in mathematics because it allows us to work with expressions more easily.When given an expression like \( \frac{1}{(-27)^{1/3}} \), where we already know that \((-27)^{1/3}\) simplifies to -3, the next step is to simplify the whole expression further:

• The expression becomes \(\frac{1}{-3}\).
• This can be rewritten as \(-\frac{1}{3}\), which is its simplest form.

Simplification may involve reducing fractions, cancelling out terms, or restructuring an expression to highlight its simplest most workable form. Simplifying expressions removes complexity and often reveals what the problem is really asking or how easily it can be solved. It's the final touch in reaching an answer that is not only correct but also clear.