Problem 7 - Algebra A Combined Function

Question

Use radical notation to rewrite each expression. Simplify if possible. $$ 169^{1 / 2} $$

Step-by-Step Solution

Answer

The expression \(169^{1/2}\) simplifies to 13.

Step 1: Understanding Radical Notation

To express an expression using radical notation, we need to convert it from exponential form to radical form. The expression given is \(169^{1/2}\). In radical notation, \(a^{1/n} = \sqrt[n]{a}\). Therefore, \(169^{1/2}\) can be expressed as \(\sqrt{169}\). We have rewritten the expression using the radical sign.

Step 2: Simplify the Radical Expression

Now, let's simplify \(\sqrt{169}\). To simplify, we find the number when multiplied by itself gives 169. 13 is the number such that \(13 \times 13 = 169\). Therefore, \(\sqrt{169} = 13\). Hence, the expression \(169^{1/2}\) simplifies to 13.

Key Concepts

Exponential Form

In mathematics, exponential form is a way of expressing numbers using a base and an exponent. The general idea is that an exponent indicates how many times the base is multiplied by itself. For example, in the expression \(169^{1/2}\), 169 is the base and \(1/2\) is the exponent.

The notation \(a^{n/m}\), where \(a\) is the base, \(n\) and \(m\) are integers, signifies that "\(a\) raised to the power of \(n/m\)." Many times, getting to grips with exponential form involves recognizing these expressions and knowing how to convert them into other forms, like radical form.

• If the exponent is a fraction like \(1/2\), this translates to the square root in radical form.
• The numerator of the fraction is the power we raise the result to, and the denominator is the root.

So, \(169^{1/2}\) is equivalent to \(\sqrt{169}\), demonstrating how to leap from exponential to radical forms.

Simplifying Radicals

Simplifying radicals is the process of finding the simplest form of a radical expression. When you simplify a radical, you are converting it to a form where no further factor can be taken out of the root. This generally makes the expression easier to work with and understand.

For instance, consider \(\sqrt{169}\). To simplify it, we search for the integer that, when multiplied by itself, equals 169. This means identifying if 169 is a perfect square. Since 13 times 13 equals 169, we know that \(\sqrt{169} = 13\).

Here’s a quick way to think about simplifying radicals:

• Find if the number is a perfect square, cube, or higher roots, depending on the radical.
• If it’s a perfect square, like 169, extract the root totally to simplify.

Simplifying such expressions can streamline your computation processes, especially in more complex algebraic equations.

Radical Expressions

Radical expressions are mathematical expressions that contain a radical symbol with a radicand underneath. They are usually used to denote roots like square roots, cube roots, etc.

The radical symbol \(\sqrt{}\) represents the square root, while \(\sqrt[n]{}\) denotes the "n-th" root; hence, the general process of manipulating these expressions is a frequent necessity in algebra.

Converting from and to radical expressions can be useful:

• It assists in simplifying complex expressions.
• Underlying relationships tend to be more visible when expressed in radical notation.

To enhance your understanding of radicals:

Practice converting between radical and exponential forms.

Engage in simplifying radical expressions to understand the foundation of solving quadratic equations better.

Understanding how to manipulate radical forms is essential in various algebraic contexts and helps in both geometry and calculus.