Problem 11 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rewrite the number without radicals or exponents.. $$ (-8)^{2 / 3} $$

Step-by-Step Solution

Answer

The simplified expression without radicals or exponents for \((-8)^{\frac{2}{3}}\) is 4.

Step 1: Evaluate the cube root

: We begin with calculating the cube root of the base (-8). The cube root of a number x is the number that, when multiplied by itself three times, gives the original number x. In mathematical notation, the cube root of x is written as \(x^{1/3}\). So, we have: \[ (-8)^{\frac{1}{3}} \] Since \((-2) * (-2) * (-2) = -8\), the cube root of -8 is -2: \[ (-8)^{\frac{1}{3}} = -2 \]

Step 2: Square the result

: Now, we need to square the result obtained in Step 1, which gives us: \[ ((-8)^{\frac{1}{3}})^{2} = (-2)^{2} \] When squaring -2, we multiply -2 by itself: \[ (-2)^{2} = 4 \]

Step 3: Final answer

: Thus, the expression \(((-8)^{\frac{2}{3}}\)) can be rewritten without radicals or exponents as: \[ (-8)^{\frac{2}{3}} = 4 \] So, the simplified expression is 4.

Key Concepts

Cube Root

The cube root is an essential mathematical concept that involves finding a number which, when multiplied by itself twice, results in the original number. It is represented as \(x^{1/3}\). Unlike square roots, cube roots can be taken of negative numbers since a negative number multiplied by itself three times will still result in a negative product. For example, in our exercise, we find the cube root of \(-8\). To verify: \((-2) \times (-2) \times (-2) = -8\). As seen, \(-2\) is the cube root of \(-8\).
Understanding cube roots helps in various calculations, especially when dealing with non-linear equations and real-world problems where volume calculations are essential. Cube roots are crucial in simplifying expressions and solving equations.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number to a power. It is denoted by \(x^n\), where \(x\) is the base and \(n\) is the exponent. This operation represents the number of times the base is multiplied by itself. In our example, we are dealing with \((-8)^{2/3}\), which involves a fractional exponent.

• The numerator of the fraction (2) indicates squaring.
• The denominator (3) indicates taking a cube root.

This expression highlights a key aspect of exponentiation: fractional exponents combine roots and powers. Thus, \((-8)^{2/3}\) means the cube root of \(-8\) squared, which simplifies through step-by-step operations.
Exponentiation is widely used in science, engineering, and financial mathematics. Mastering this concept can simplify complex problems and streamline calculations.

Simplifying Expressions

Simplifying expressions involves breaking down complex mathematical expressions into simpler, more manageable forms. In our scenario, the expression \((-8)^{2/3}\) was simplified by first identifying each operation governed by the exponent's fraction.
The process included:

• Calculating the cube root of \(-8\), resulting in \(-2\).
• Squaring \(-2\) to achieve the final simplified form.

Through simplification, the expression \((-8)^{2/3}\) becomes \(4\) after performing these operations.
Simplifying expressions is a crucial skill in algebra that helps to clear complex expressions, making them readily understandable and easier to solve. It is vital in everyday problem-solving and advanced mathematics.