Problem 23 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ (-8)^{2 / 3} $$

Step-by-Step Solution

Answer

The value of \((-8)^{2/3}\) is 4.

Step 1: Understand the Expression

We have the expression \[(-8)^{2/3}\]This means we're finding the cube root of -8 and then squaring the result.

Step 2: Find the Cube Root

To find the cube root of -8, we determine the number that, when cubed, gives -8. The cube root of -8 is -2, since \((-2)^3 = -8\).

Step 3: Square the Result

Next, we square the result from Step 2. Squaring -2 gives \((-2)^2 = 4\). So, \((-8)^{2/3} = 4\).

Key Concepts

Cube Root

The idea of cube roots is all about reversing the process of cubing a number. When you see the cube root symbol, it’s asking: what number, when multiplied by itself three times, results in the given number? For instance, finding the cube root of -8 involves asking what number, when used in a multiplication three times, equals -8.
- In this exercise, the cube root of -8 was found to be -2 because \[(-2) \times (-2) \times (-2) = -8.\]
- Notably, cube roots can handle negative numbers easily. This is different from square roots, where the operation is not defined for negative inputs in the set of real numbers.
Cube roots cover both positive and negative results as they maintain the sign of the original number. In simple terms, if you start with a negative and cube it, the result remains negative.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, a base, and an exponent. The exponent tells you how many times to use the base in a multiplication. For example, in \[a^b\]"a" is the base, and "b" is the exponent.
- When we see fractional exponents like \[(-8)^{2/3}\]it implies first taking a root of the base number and then raising that result to a power. Here, "2/3" means finding the cube root and then squaring the result.
- Fractional exponents are shortcuts that link roots and powers together, allowing us to work out complex expressions systematically.
Understanding how to manipulate exponents is crucial for simplifying expressions and solving equations in algebra.

Negative Bases

Negative bases can produce surprising results when raised to certain powers. These powers can be integer or rational numbers, and each influences outcomes differently.
- Negative bases raised to an even power result in positive products because two negatives \[multiplied\] together yield a positive. For example, \[(-2)^2 = 4.\]
- When dealing with fractional exponents, like \[2/3\], the expression first involves finding a root (like the cube root) and then raising the result to a power.
- Managing negative bases requires careful attention to the rules of exponents to correctly evaluate the sign and magnitude of results. This is key, especially when dealing with tasks like our example, \[(-8)^{2/3}.\]
Mastering how negative bases work with various powers aids in simplifying expressions and achieving correct solutions in mathematics.