Problem 2 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rewrite the number without radicals or exponents.. $$ \sqrt[3]{-27} $$

Step-by-Step Solution

Answer

The given expression, \(\sqrt[3]{-27}\), can be rewritten without radicals or exponents as -3, since \((-3)^3 = -27\).

Step 1: Identify the root

In this case, we are given the cubic root of -27, which can be written as \(\sqrt[3]{-27}\).

Step 2: Find the cubic root

To find the cubic root of -27, we need to think of a number that, when multiplied by itself three times (i.e., raised to the power of 3), results in -27. Recall that the product of an odd number of negative numbers is also a negative number. After some trial and error, we find that -3 raised to the power of 3 gives -27: \((-3)^3 = -3 \times -3 \times -3 = -27\)

Step 3: Rewrite the expression without radicals or exponents

Since we found that the cubic root of -27 is -3, we can rewrite the original expression \(\sqrt[3]{-27}\) without radicals or exponents as simply -3.

Key Concepts

Radicals

A radical, often represented by the symbol \(\sqrt{}\), is a way to express the root of a number. Radicals allow us to find numbers that, when raised to a certain power, give the original number. In this context, the expression \(\sqrt[3]{-27}\) represents the cube root of -27.

• Cubic Roots: A cubic root asks, "What number, when raised to the third power, equals the given number?" For \(-27\), we are finding a value \(x\) such that \(x^3 = -27\).
• Symbol: The number inside the radical, known as the radicand, is the number you want to find the root of.For the cube root of -27, the radicand is \(-27\).
• Usage: Radicals can be used with numbers beyond cube roots, like square roots (\(\sqrt{}\)) or fourth roots (\(\sqrt[4]{}\)). They allow powerful simplifications in mathematics.

Understanding radicals inside mathematical expressions helps clarify complex calculations and how numbers relate to their roots.

Exponents

Exponents are mathematical notations indicating the number of times a number, called the base, is multiplied by itself. They are written as a small number to the top right of the base, like \( x^3 \) where 3 is the exponent.

• Cubic Exponents: When the exponent is 3, the operation is termed 'cubing.' It means multiplying the number by itself three times: \(x^3 = x \times x \times x\).
• Relationship with Roots: Exponents and radicals are opposite operations. For instance, finding a cube root is the reverse process of cubing \( (\sqrt[3]{x} = x^{1/3}) \).
• Negative Bases: When the base is a negative number, negative results can be found, especially when raised to an odd exponent. In our example, \((-3)^3 = -27\).

Exponents help us express repeated multiplication succinctly and are crucial for simplifying expressions and solving equations.

Negative Numbers

Negative numbers are values less than zero, positioned to the left of zero on a number line. They often represent opposites in real-world scenarios, like debt or temperatures below zero.

• Properties: Multiplying an even number of negative numbers results in a positive number. Multiplying an odd number of negative numbers results in a negative number, which is crucial for understanding cubic roots.
• Cubic Roots of Negatives: For cube roots, a negative number, when multiplied by itself three times, remains negative. So, the cube root of \(-27\) is \(-3\).
• Example Use: Consider a debt of 27 dollars as a negative 27. When split equally into three parts, each part will be \(-3\), representing the individual debts.

Negative numbers extend our understanding of mathematics to include values and scenarios less than zero, helping solve a wider range of problems.