Problem 42 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ (-27)^{-2 / 3} $$

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{1}{9}\).

Step 1: Understand the Expression

We need to evaluate the expression \((-27)^{-2/3}\). The exponent \(-2/3\) consists of a negative sign and a fraction, indicating both an inverse operation and a root operation.

Step 2: Address the Negative Exponent

The negative exponent \(-2/3\) means we need to take the reciprocal of the base raised to the positive exponent. Thus, \((-27)^{-2/3} = \frac{1}{(-27)^{2/3}}\).

Step 3: Simplify the Fractional Exponent

The fractional exponent \(2/3\) indicates that we first take the cube root and then square the result. Thus, \((-27)^{2/3} = ((-27)^{1/3})^2\).

Step 4: Calculate the Cube Root

Calculate \((-27)^{1/3}\), which is the cube root of -27. The cube root of -27 is -3, because \((-3) imes (-3) imes (-3) = -27\).

Step 5: Square the Result

Now square the cube root result: \((-3)^2 = 9\). This means \((-27)^{2/3} = 9\).

Step 6: Evaluate the Final Expression

Returning to the expression, we have \(\frac{1}{9}\) because \((-27)^{-2/3} = \frac{1}{(-27)^{2/3}} = \frac{1}{9}\).

Key Concepts

Negative Exponents

Negative exponents can sometimes seem intimidating, but they're quite straightforward once you learn the rules. When you see a negative exponent like in \((-27)^{-2/3}\), think of it as an instruction to flip the number - to find the reciprocal.
A reciprocal simply means one over the number.
- For example, if you have \(a^{-n}\), it means \(\frac{1}{a^n}\).
So, in the exercise, \((-27)^{-2/3}\) becomes \(\frac{1}{(-27)^{2/3}}\).
This step is crucial because it transforms the problem from dealing with negative powers to handling positive powers, making it easier to manage further calculations.

Reciprocal

The reciprocal of a number is simply that number flipped.
For example, the reciprocal of 5 is \(\frac{1}{5}\).
When working with exponents, understanding reciprocals helps when you've got negative exponents.
This is because a negative exponent directly indicates using a reciprocal.
Consider the expression \((-27)^{-2/3}\). Here we needed the reciprocal due to the negative exponent:
- Thus, \((-27)^{-2/3} = \frac{1}{(-27)^{2/3}}\).
Now, instead of focusing on a negative exponent, we shift our attention to solving \(\frac{1}{(-27)^{2/3}}\) with positive exponents.
This technique makes calculations easier and often tidier.

Cube Root

Calculating a cube root may sound complex, but it's simpler than it seems.
The cube root of a number is another number which, when multiplied by itself three times, gives the original number.
In our case for \((-27)\), we looked for what number multiplied three times results in \(-27\).
- It turns out this number is \(-3\) since \((-3) \times (-3) \times (-3) = -27\).
Finding cube roots acts as the first step in simplifying fractional exponents like \(\frac{2}{3}\).
In the expression \((-27)^{2/3}\), the cube root is calculated first, leading to the next step of handling the exponent.

Square

Squaring a number means multiplying it by itself.
Once we found the cube root, which is \(-3\), the exercise required us to square \(-3\) to move on with the calculation.
- When you square \(-3\), the negative sign is also factored in: \((-3) \times (-3) = 9\).
This step often concludes operations with fractional exponents like \(\frac{2}{3}\), where the second part of the calculation is squaring the initial result.
It brings our answer one step closer to completion, allowing us then to find the reciprocal for a final answer if necessary.