Problem 1 - Algebra A Combined Function

Question

Use radical notation to rewrite each expression. Simplify if possible. $$ 49^{1 / 2} $$

Step-by-Step Solution

Answer

The expression \(49^{1/2}\) simplifies to 7.

Step 1: Identify the Radical Notation

The expression given is in fractional exponent form: \[49^{1/2}\]A fractional exponent in the form \(n^{1/m}\) can be rewritten in radical notation as \(\sqrt[m]{n}\). Here, \(m = 2\), so we have a square root.

Step 2: Convert to Radical Notation

Rewrite the expression using radical notation. Since \(49^{1/2}\) is equal to \(\sqrt{49}\), the expression becomes:\[\sqrt{49}\]

Step 3: Simplify the Radical Expression

Now, find the square root of 49. We know that 7 multiplied by itself equals 49:\[7 \times 7 = 49\]Thus, the simplified form is:\[\sqrt{49} = 7\]

Key Concepts

Fractional Exponents

Did you know that fractional exponents are just another way of expressing roots? For example, the expression \( 49^{1/2} \) might look a bit puzzling at first. But once you understand how these exponents work, you'll find them less daunting. The key point to understand is that fractional exponents can represent roots:

• In general, an exponent in the form \( n^{1/m} \) signifies the \( m \)-th root of \( n \).
• Specifically, when \( m = 2 \), as in \( 49^{1/2} \), you're looking at the square root of \( 49 \).

To convert this fractional exponent back into something more familiar, you can rewrite \( n^{1/m} \) using radical notation. For example, \( 49^{1/2} \) is equivalent to \( \sqrt{49} \). Understanding this relationship can make handling problems involving roots much simpler!
Remember that converting between these forms is vital to simplifying expressions and solving equations involving powers and roots.

Square Root

Square roots are some of the most common roots you'll encounter in algebra. They are simple yet powerful tools in simplifying expressions. The square root of a number \( n \) is a value that, when multiplied by itself, gives \( n \). For instance, the square root of \( 49 \) is \( 7 \) because \( 7 \times 7 = 49 \).

• The symbol for square root is the radical symbol \( \sqrt{} \).
• \( \sqrt{n} \) represents the square root and implies finding a number that, squared, results in \( n \).

Understanding square roots can also be useful in solving quadratic equations or analyzing graphs. Importantly, any positive number has two square roots: one positive and one negative. However, by convention, \( \sqrt{n} \) refers to the principal square root, which is the non-negative one. For example, \( \sqrt{49} = 7 \).

Simplifying Radicals

Simplifying a radical is the process of rewriting it in its simplest form. It's important to recognize perfect squares, as they allow you to simplify square roots directly.

• For instance, with \( \sqrt{49} \), since \( 49 \) is a perfect square of \( 7 \), the radical simplifies directly to \( 7 \).
• When dealing with \( \sqrt{n} \), where \( n \) is not a perfect square, you can factor \( n \) into its prime components to simplify.

For example, if faced with a more complicated expression like \( \sqrt{72} \), you would factor \( 72 \) into \( 36 \times 2 \), knowing \( 36 \) is a perfect square. Thus, \( \sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2} \). Always watch for these opportunities to simplify, as they'll help you solve problems more efficiently and reveal more about the numbers and expressions you're working with.