Problem 80 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{(b+7)^{8}(b-7)^{6}} $$

Step-by-Step Solution

Answer

Question: Simplify the radical expression: \(\sqrt{(b+7)^{8}(b-7)^{6}}\) Answer: \((b+7)^{4}(b-7)^{3}\)

Step 1: 1. Identify the properties of exponents and radicals

We need to use the property \(\sqrt{x^2}=x\) when the exponent is even and the property \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\) so we can distribute the square root to each term inside the parenthesis.

Step 2: 2. Apply the rules to the given expression

Now, let's apply these rules to the given expression: $$ \sqrt{(b+7)^{8}(b-7)^{6}} $$ According to the properties, we can distribute the square root to each term: $$ \sqrt{(b+7)^{8}} \cdot \sqrt{(b-7)^{6}} $$

Step 3: 3. Simplify the square root of each term

Now, for each term with an even power, we can find its square root: $$ (b+7)^{8/2} \cdot (b-7)^{6/2} $$

Step 4: 4. Simplify the powers

Now, let's simplify the exponents: $$ (b+7)^{4} \cdot (b-7)^{3} $$ So, the simplified expression is: $$ (b+7)^{4}(b-7)^{3} $$

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial when you're simplifying expressions with exponents. Exponents tell us how many times to multiply a base number by itself. There are several useful properties that help us in simplifying expressions:

• Product of Powers: This rule states that when multiplying similar bases, you add their exponents. For example, \(a^m \cdot a^n = a^{m+n}\).
• Power of a Power: This property indicates that when raising a power to another power, you multiply the exponents, e.g., \( (a^m)^n = a^{m \cdot n}\).
• Quotient of Powers: When dividing similar bases, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\). This is particularly useful when you encounter fractions in radical expressions.
• Power of a Product: This property is used when a product is raised to a power, allowing distribution of the exponent to each factor: \( (ab)^n = a^n \cdot b^n\).

These properties are instrumental when working with roots and radicals as they help break down complex expressions into simpler ones.

Radical Expressions

Radical expressions involve roots of numbers or variables. The most common radical is the square root, but there are other types like cube roots and fourth roots. Radicals can be tricky if you're not familiar with their properties.

• Square Root: The square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\). It's denoted by \(\sqrt{x}\).
• Radical Index: The small number above the root, \(\sqrt[n]{x}\), indicates the type of root. If there's no number, it's usually a square root.
• Multiplying Radicals: The product property of radicals allows you to multiply roots: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\).
• Quotient of Radicals: You can divide radicals using \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\).

In the exercise, you first break down the radical into two parts using the property that you can distribute \(\sqrt{}\) over multiplication. Then each part's radical is simplified using the properties of the exponents.

Simplification of Expressions

Simplifying radical expressions often involves combining the properties of exponents and radicals. This process makes the expression easier to work with. Let's simplify using the example:

• First, you'll use the property \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\) to separate terms inside a square root. In our exercise, \(\sqrt{(b+7)^8 \cdot (b-7)^6}\) is broken down to \(\sqrt{(b+7)^8} \cdot \sqrt{(b-7)^6}\).
• Next is realizing even exponents allow perfectly simplified roots. For instance, \(\sqrt{(b+7)^8}\) simplifies to \( (b+7)^{8/2} = (b+7)^4\).
• Finally, simplify each term individually using the power rule. \(\sqrt{(b-7)^6} = (b-7)^{6/2} = (b-7)^3\).

This step-by-step breakdown transforms the original expression to a much simpler form. Mastering simplification is vital in math as it strengthens problem-solving skills.