Problem 48 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left(-\frac{1}{8}\right)^{-5 / 3} $$

Step-by-Step Solution

Answer

The value of \( \left(-\frac{1}{8}\right)^{-5/3} \) is \(-32\).

Step 1: Understand the Negative Exponent

A negative exponent indicates that we will be taking the reciprocal of the base. So, for the expression \( \left(-\frac{1}{8}\right)^{-5/3} \), we first interpret the negative sign in the exponent by taking the reciprocal of \(-\frac{1}{8}\). This gives us \(-8\) raised to the power of \(5/3\).

Step 2: Simplify the Fractional Exponent

The fractional exponent \(5/3\) implies two operations: a cube root (denominator) and raising to a power (numerator). Begin by taking the cube root of \(-8\). Since \( (-2)^3 = -8 \), the cube root of \(-8\) is \(-2\). Thus, we rewrite the expression as \((-2)^5\).

Step 3: Calculate the Power

Now calculate \((-2)^5\). Raise \(-2\) to the 5th power: \((-2) \times (-2) \times (-2) \times (-2) \times (-2) = -32\).

Step 4: Combine the Results

Having computed \((-2)^5\) as \(-32\), we find that \( \left(-\frac{1}{8}\right)^{-5/3} = -32 \).

Key Concepts

Negative Exponents

Negative exponents might initially seem confusing, but they are quite simple once you get the hang of them. The negative sign in an exponent indicates that we need to take the reciprocal of the base. Consider the expression \((a^{-n})\). The negative exponent \(-n\) tells us to flip \(a\), making it \(\frac{1}{a^n}\). So, \(a^{-n} = \frac{1}{a^n}\).

For example, in the exercise \(\left(-\frac{1}{8}\right)^{-5/3}\), you start by considering the negative exponent \(-5/3\). This requires taking the reciprocal of \(-\frac{1}{8}\). So, it becomes \(-8\) raised to the power of \(5/3\).

Negative exponents are a great tool for simplifying problems, allowing for more flexible and precise mathematical operations. Remember, flipping the base alleviates the negative exponent.

Fractional Exponents

Fractional exponents can be a bit tricky at first. They combine two operations: roots and powers. The numerator indicates the power to which the number is raised, while the denominator tells us which root to take. If we see \(a^{m/n}\), we interpret it as \((\sqrt[n]{a})^m\) or \(\sqrt[n]{a^m}\). Both interpretations yield the same result but might simplify the problem differently.

In the exercise, \((-8)^{5/3}\) involves a fractional exponent of \(5/3\). Here, the denominator \(3\) specifies that we first find the cube root of \(-8\). This equals \(-2\) as \((-2)^3 = -8\). We then raise the result to the power of \(5\) to finish the calculation.

Fractional exponents can simplify expressions by representing both roots and powers in a single compact form, making calculations more manageable and less cluttered.

Cube Roots

Cube roots relate closely to solving problems involving radicals. The cube root of a number \(a\) is a value \(b\) such that \(b^3 = a\). Cube roots are used to balance equations and simplify expressions in mathematics. They are particularly useful for large numbers or when dealing with fractional exponents as we saw earlier.

In the given exercise \(\left(-\frac{1}{8}\right)^{-5/3}\), when the expression is transformed to \((-8)^{5/3}\), the challenge becomes finding \(\sqrt[3]{-8}\). Negative numbers can have real cube roots; for example, the cube root of \(-8\) is \(-2\) because \((-2)^3 = -8\).

• Cube roots simplify calculations involving three identical factors.
• They help in transforming radical expressions into a form involving whole numbers or rational numbers.

By finding the cube root first, we significantly simplify our calculations for expressions with multiple operations.