Problem 47 - Algebra 2 and Trigonometry

Question

In \(38-57,\) write each radical expression as a power with positive exponents and express the answer in simplest form. The variables are positive numbers. $$ \sqrt{49 x^{2}} $$

Step-by-Step Solution

Answer

The simplest form of \(\sqrt{49 x^{2}}\) is \(7x\).

Step 1: Recognize the Square Root

The exercise gives us the expression \(\sqrt{49x^2}\). A square root is an expression that can be rewritten using exponents. Specifically, \(\sqrt{a} = a^{1/2}\). This is the key transformation needed to convert a radical into a power.

Step 2: Apply the Power Rule to the Square Root

Using the rule from the previous step, rewrite \(\sqrt{49x^2}\) as \((49x^2)^{1/2}\). This transformation allows us to further manipulate the expression using the laws of exponents.

Step 3: Apply Exponent Distributive Property

The exponent distributive property states that \((ab)^n = a^n \cdot b^n\). Apply this rule to \((49x^2)^{1/2}\) by breaking it into \(49^{1/2} \cdot (x^2)^{1/2}\).

Step 4: Simplify Each Term

Simplify \(49^{1/2}\) and \((x^2)^{1/2}\): \(49^{1/2} = 7\) because \(7^2 = 49\), and \((x^2)^{1/2} = x^{2 \cdot 1/2} = x^1 = x\).

Step 5: Combine the Simplified Terms

The expression now is \(7\cdot x\), which simplifies to just \(7x\). This is the expression written as a power with positive exponents and in simplest form.

Key Concepts

Radical Expressions

Let's start with radical expressions, which are mathematical notations used to indicate roots of numbers or expressions. The most common radical is the square root, but they can also extend to cube roots, fourth roots, and so on. When you see a radical symbol, such as \(\sqrt{}\), it signifies the square root of the enclosed expression. If you're dealing with cube roots, you'd use \(\sqrt[3]{}\), and this pattern continues for higher roots. These expressions allow us to represent numbers that are not perfect squares - numbers that don't have an exact whole number as their square root.

Key properties of radicals are essential to manipulate expressions and solve equations:

• The expression \(\sqrt{a}\) can be viewed as \(a^{1/2}\), and similarly, the cube root \(\sqrt[3]{a}\) is \(a^{1/3}\).
• To help simplify radicals, remember that multiplying radical expressions follows the property \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).
• When simplifying radicals, focus on finding the largest perfect square factor of the number under the radical for squares, or the largest perfect cube factor for cubes.

Square Root

The square root is a specific case of radical expressions, typically denoted by the symbol \(\sqrt{}\). When we talk about finding the square root of a number, we're essentially looking for a value that, when multiplied by itself, gives the original number back. For example, the square root of 49 is 7 because \(7 \times 7 = 49\).

Understanding the square root also involves considering its fundamental property: \(\sqrt{a} = a^{1/2}\). This connection lets us rewrite expressions involving square roots into exponential form, making them easier to manipulate using laws of exponents.

• In solving equations, square roots can be simplified by recognizing perfect squares from numbers. For instance, \(\sqrt{49} = 7\) and \(\sqrt{x^2} = x\), assuming \(x\) is positive, to keep things simple.
• Keep in mind that the square root operation can extend to algebraic expressions, such as \(\sqrt{x^2}\), where it equals \(x\), representing the absolute value since squares negate any inherent negative sign.

Laws of Exponents

The laws of exponents are crucial for understanding how to manipulate expressions with exponents, including those derived from radicals. Applying these rules can simplify complex expressions and solve various mathematical problems. Here are some key laws that come into play with the transformation of radicals:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\). This rule allows multiplying together like bases.
• Power of a Power: \((a^m)^n = a^{m \cdot n}\). This is especially useful when dealing with expressions like \((x^2)^{1/2}\), which simplifies to \(x^{2 \cdot 1/2} = x^1 = x\).
• Power of a Product: \((ab)^n = a^n \cdot b^n\). This law is handy to distribute exponents across a product inside parentheses, just like \((49x^2)^{1/2} = 49^{1/2} \cdot (x^2)^{1/2}\).

Understanding these laws helps break down expressions into simpler forms, making them easier to handle and solve. By mastering the laws of exponents, you'll be better equipped to convert between radical and exponential forms with ease.