Problem 23 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

4

Step 1: Understand the Expression

The given expression is \((-8)^{2/3}\). The base is -8 and the exponent is a fraction \(\frac{2}{3}\), indicating both a root and a power.

Step 2: Simplify the Base Using Roots and Powers

A fractional exponent \(\frac{2}{3}\) can be broken down as a root and a power. This means \((-8)^{2/3} = \left((-8)^{1/3}\right)^2\). We need to find the cube root of -8 first.

Step 3: Calculate the Cube Root

The cube root of -8 is the value that, when multiplied by itself three times, gives -8. Since \((-2) \times (-2) \times (-2) = -8\), the cube root of -8 is -2. So, \((-8)^{1/3} = -2\).

Step 4: Raise the Result to the Power of 2

Now, we need to calculate \((-2)^2\). This means multiplying -2 by itself: \((-2) \times (-2) = 4\).

Step 5: Combine the Steps

Putting it all together: \((-8)^{2/3} = ((-8)^{1/3})^2 = (-2)^2 = 4\). The value of the expression \((-8)^{2/3}\) is 4.

Key Concepts

Cube Root

The concept of cube roots is essential to understanding fractional exponents. A cube root is the number that must be multiplied by itself three times to yield the original number.
For example, the cube root of -8 is -2 because when you multiply -2 by itself twice, and then once more, the result is -8:

• First Multiplication: ewline egin{equation} -2 imes -2 = 4 ewline ewline ewline Second Multiplication: ewline 4 imes -2 = -8 ewline ewline ewline ext{Result: } -8 ewline ewline ewline ext{so } ext{Cube root of } -8 = -2 ewline ewline ewline onumberewline ewline ewline ight) ewline ight) ight) ewline The cube root is indicated by the radical sign with an index of 3, or it can be expressed using a fractional exponent, like so:
  • \((-8)^{1/3} \).This property of cube roots extends to any number, and recognizing these basic patterns makes it easier to handle fractional exponents.

Exponents and Powers

Exponents and powers are fundamental in algebra and represent how many times a number is multiplied by itself. When we see an expression like \\((-8)^{2/3}\),\ the fraction suggests two operations: taking the cube root (denominator) and squaring it (numerator). Here's how you break it down:

• Begin with the cube root: \((-8)^{1/3} = -2\).
• Then square the result: \((-2)^2 = 4\).
• Hence, \((-8)^{2/3} = 4\).

Understanding these steps helps when the base is negative because exponent rules guide the operations. Additionally, knowing when to take roots and raise powers gives clarity to complex expressions.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form. With fractional exponents, this means using known properties of exponents and roots.
The original expression \\((-8)^{2/3}\) might initially look complicated. By breaking it into manageable parts, it becomes easier to handle.

• Start by dealing with the fractional component separately.
  • Cube root: \((-8)^{1/3} = -2\)
  • Square the outcome: \((-2)^2 = 4\)
  • This simplifies the original expression to \(4\).
• Using these steps helps eliminate complexity, resulting in an answer that is simple and straightforward. Recognizing patterns in numbers and exponents ensures efficiency.