Problem 32 - Intermediate Algebra

Question

Write with positive exponents. Simplify if possible. $$ (-8)^{-4 / 3} $$

Step-by-Step Solution

Answer

The positive exponent form is \(\frac{1}{16}\).

Step 1: Understand Negative Exponents

The expression involves a negative exponent \((-8)^{-4/3}\). A negative exponent indicates that we take the reciprocal of the base, which means \(a^{-n} = \frac{1}{a^n}\). Thus, \((-8)^{-4/3} = \frac{1}{(-8)^{4/3}}\).

Step 2: Apply the Fractional Exponent

A fractional exponent like \(\frac{4}{3}\) implies both a root and a power. Specifically, \(a^{m/n} = \sqrt[n]{a^m}\). Thus, \((-8)^{4/3} = \left(\sqrt[3]{-8}\right)^4\).

Step 3: Evaluate the Cube Root

Calculate \(\sqrt[3]{-8}\). Since \(-8 = (-2)^3\), the cube root is \(\sqrt[3]{-8} = -2\).

Step 4: Raise to the Power of 4

Now raise the cube root result to the power of 4: \((-2)^4 = 16\).

Step 5: Take the Reciprocal

We earlier took the reciprocal in the negative exponent step, so the final answer is \(\frac{1}{16}\). This value was obtained as \((-8)^{-4/3} = \frac{1}{16}\).

Key Concepts

Fractional Exponents

Fractional exponents can be a little tricky, but they are basically a way to express both roots and powers in one symbol. When you see an exponent like \( \frac{m}{n} \), this means that you should take the \( n \)-th root of a number and then raise the result to the \( m \)-th power. This is written mathematically as \( a^{m/n} = (\sqrt[n]{a})^m \).
In the case of our exercise, \((-8)^{-4/3}\), the exponent \( \frac{4}{3} \) includes both a cube root and a power of 4. This means we will first find the cube root of \(-8\) and then raise the result to the fourth power. Knowing how to handle fractional exponents is crucial, as it can help simplify complex expressions involving roots and powers.

Simplifying Expressions

Simplifying expressions is all about making them easier to work with. This involves breaking down complicated expressions to their simplest form. For expressions with exponents, simplification often means applying exponent rules such as the product of powers, quotient of powers, and power of a power.
In our example of \((-8)^{-4/3}\):

• The negative exponent tells us to take the reciprocal, turning it into \( \frac{1}{(-8)^{4/3}} \).
• The fractional exponent \( \frac{4}{3} \) means both taking a cube root and raising to a power.
• By simplifying step-by-step, we calculate \((-8)^{4/3}\) to be \(16\), and therefore the expression simplifies to \( \frac{1}{16} \).

Understanding these simplification steps is key to solving similar problems in a breeze.

Roots and Powers

Roots and powers are fundamental concepts in algebra that let us express numbers compactly. They allow us to handle very large or very small numbers conveniently.
The root part of \((\sqrt[n]{a})\) represents a number that, when multiplied by itself \( n \) times, yields \( a \). A common example is the square root \( \sqrt{a} \), where the number multiplied by itself twice gives \( a \).
Powers, on the other hand, involve multiplying a base by itself a certain number of times, shown as \( a^m \). So \((-2)^4 = 16\) represents multiplying \(-2\) by itself four times, resulting in 16.
Combining both roots and powers, such as in fractional exponents, adds versatility in expressions and can simplify complicated calculations. Understanding how to use roots and powers effectively can help you tackle advanced algebra problems with more confidence.