Problem 42 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

\((-27)^{-2/3} = \frac{1}{9}\).

Step 1: Understand the Exponent

The expression \[(-27)^{-2/3}\]has a negative exponent and a fractional exponent: -2/3. The negative signifies reciprocal, and the fraction indicates a root and power operation.

Step 2: Convert Negative Exponent

The negative exponent -2/3 can be converted to a positive exponent by taking the reciprocal of the base:\[(-27)^{-2/3} = \frac{1}{(-27)^{2/3}}\]

Step 3: Break Down the Fractional Exponent

The fractional exponent \(2/3\) can be broken down into a root and a power. This means to first take the cube root (as indicated by the denominator) and then square the result (as indicated by the numerator).

Step 4: Find the Cube Root

Find the cube root of -27. Since we know that \((-3)^3 = -27\), the cube root of -27 is -3.

Step 5: Square the Result of the Cube Root

Now, square the result of the cube root:\[(-3)^2 = 9\]

Step 6: Calculate the Reciprocal

Finally, find the reciprocal of the value obtained in the previous step to account for the original negative exponent:\[\frac{1}{9}\]

Key Concepts

Negative exponents

When you encounter a negative exponent, it might seem a bit tricky at first. However, it's a simple concept that revolves around the idea of a reciprocal. A negative exponent like \(-a\) indicates that instead of multiplying the base by itself, you should take the reciprocal. For instance, \(x^{-a} = \frac{1}{x^a}\). This means that the base, instead of being raised to a positive power, is put in the denominator with the exponent turned positive.
By thinking of negative exponents in terms of reciprocals, they become much easier to handle.

• They allow us to rewrite expressions as fractions.
• Negative exponents are useful in simplifying expressions and solving equations.

Fractional exponents

Fractional exponents are a way of denoting roots along with powers in a compact form. The expression \(x^{m/n}\) means you first take the \(n\)-th root of \(x\) and then raise it to the \(m\)-th power.
These exponents combine root and power operations into one notation, making it easier to manage and interpret mathematical expressions.

• The denominator of the fraction indicates the type of root.
• The numerator shows the power to which the result is raised.

For example, in the expression \((-27)^{2/3}\), the cube root is taken first, followed by squaring the result.

Reciprocal

The reciprocal of a number is what you multiply by to get 1. If you have a number \(x\), its reciprocal is \(\frac{1}{x}\). This concept frequently appears when dealing with negative exponents, as noted earlier.
Utilizing reciprocals requires switching the numerator and denominator of a fraction, essentially "flipping" it over.

• To find a reciprocal, divide 1 by the number.
• Multiplying a number by its reciprocal always gives 1.

In the example \[\frac{1}{9}\], we found the reciprocal of 9, which was the outcome of previous calculations.

Cube root

The cube root of a number is a special value that, when cubed, gives the original number. For instance, if you have the number -27, its cube root is -3 because \((-3)^3 = -27\).
Cube roots can also be represented using fractional exponents, such as \(x^{1/3}\), which indicates you take the cube root of \(x\).

• Cube roots help simplify expressions involving cubes.
• Negative numbers can have real cube roots, unlike square roots.

Working with cube roots is straightforward once you recognize the relationships between cubes and their roots in both positive and negative numbers.