Problem 3 - Precalculus: Mathematics for Calculus

Question

(a) Using exponential notation, we can write \(\sqrt[3]{5}\) as _____________ . (b) Using radicals, we can write \(5^{1 / 2}\) as ____________ . (c) Is there a difference between \(\sqrt{5^{2}}\) and \((\sqrt{5})^{2} ?\) Explain.

Step-by-Step Solution

Answer

(a) \(5^{1/3}\), (b) \(\sqrt{5}\), (c) Both are equal, no difference.

Step 1: Understanding Exponential Notation

First, recognize that the cube root of a number can be expressed as an exponent. The cube root of 5, \( \sqrt[3]{5} \), can be written using exponential notation as \( 5^{1/3} \). This is because taking the cube root is equivalent to raising a number to the power of \( \frac{1}{3} \).

Step 2: Converting to Radical Form

Next, understand that a number raised to the power of \( \frac{1}{2} \) can be expressed using a square root. So, the expression \( 5^{1/2} \) is equivalent to \( \sqrt{5} \). This is because \( \frac{1}{2} \) as an exponent means taking the square root.

Step 3: Comparing Different Root Operations

Now, let's examine the expressions \( \sqrt{5^{2}} \) and \( (\sqrt{5})^{2} \). Start with \( \sqrt{5^{2}} \), which simplifies to \( \sqrt{25} = 5 \). For the second expression, \( (\sqrt{5})^{2} \) simplifies as follows: \( \sqrt{5} \) squared means multiplying \( \sqrt{5} \) by itself, resulting again in 5. Both expressions evaluate to the same value, showing that they are equivalent.

Key Concepts

Understanding Radicals

Radicals are symbols used to denote roots of numbers. The most common radical is the square root, represented by the symbol \( \sqrt{} \). This symbol indicates that we are looking for a number which, when multiplied by itself a certain number of times, results in the original number under the radical.

Some key points to remember about radicals include:

• The expression under the radical sign is called the radicand.
• The index of the radical indicates which root to take. If the index isn't visible, it's implied to be 2, such as in square roots.

Radicals can often be converted into exponential notation, which can make certain mathematical operations simpler to perform. For example, a cube root of a number \( x \) can be written as \( x^{1/3} \) in exponential form. This conversion is quite handy when dealing with expressions that are more manageable using exponent rules.

Square Root Simplifications

A square root is one of the simplest types of radicals, represented by \( \sqrt{x} \). Finding the square root of a number refers to finding what number, when multiplied by itself, equals the original number.

Let's clarify this with an example:

• The square root of 25, or \( \sqrt{25} \), is 5 because \( 5 \times 5 = 25 \).

Square roots can often be expressed in exponential notation as well. Whenever you encounter \( x^{1/2} \), it translates directly to \( \sqrt{x} \). This notation is especially useful in algebraic expressions, as it simplifies the process of solving equations by employing exponent rules.

Exploring the Cube Root

A cube root of a number indicates what number must be multiplied by itself three times to yield the original number. Mathematically, the cube root of a number \( x \) is written as \( \sqrt[3]{x} \).

Here are some practical aspects of cube roots:

• The simplest example is \( \sqrt[3]{8} \), which equals 2, because \( 2 \times 2 \times 2 = 8 \).
• In exponential form, a cube root is represented as \( x^{1/3} \).

Using exponential notation for cube roots can simplify computations, particularly when dealing with powers and logarithms. This mathematical flexibility is essential when you're simplifying expressions involving various roots, making calculations clearer and more efficient.