Problem 66 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{a^{2 n+5}} \sqrt{a^{3}} $$

Step-by-Step Solution

Answer

Answer: The product of \(\sqrt{a^{2n+5}}\) and \(\sqrt{a^3}\) is \(\sqrt{a^{2n+8}}\).

Step 1: Multiply the square roots

According to the properties of the square roots, we have: $$ \sqrt{a^{2 n+5}}\sqrt{a^3}=\sqrt{a^{2 n+5}\cdot a^3} $$

Step 2: Apply the law of exponents

To multiply the expressions inside the square root, we must apply the law of exponents. When multiplying expressions with the same base (in this case, \(a\)), you should add the exponents: $$ \sqrt{a^{2 n+5}\cdot a^3} = \sqrt{a^{(2n+5) + 3}} $$

Step 3: Simplify the expression

Now, simplify the exponent by adding the constants: $$ \sqrt{a^{(2n+5)+3}} = \sqrt{a^{2n+8}} $$ The final answer is \(\sqrt{a^{2n+8}}\).

Key Concepts

Law of Exponents

The law of exponents is a powerful tool in algebra that simplifies working with expressions involving powers. When you multiply two expressions with the same base, you simply add their exponents. For example, if you have expressions like \(a^m\) and \(a^n\), by applying the law of exponents, you find that \(a^m \cdot a^n = a^{m+n}\). This rule helps to condense the expression into a simpler form.
In the original exercise, we used this rule while working inside the square root to deal with terms \(a^{2n+5}\) and \(a^3\). Since both terms have the base \(a\), we added their exponents: \((2n+5) + 3\), resulting in \(a^{2n+8}\). It is crucial to notice that this law only applies when the bases are identical.
Understanding this principle can clarify many algebraic manipulations and lead to correctly solving problems involving powers.

Properties of Square Roots

Square roots can sometimes be confusing, but understanding their properties will help make complex expressions easier. One important property is that the square root of a product equals the product of the square roots. This means \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\). This property simplifies expressions significantly when the inside of the square root can be broken into multiple factors.
In our exercise, we saw this property in action when combining two roots: \(\sqrt{a^{2n+5}}\) and \(\sqrt{a^3}\). We merged them into a single root: \(\sqrt{a^{2n+5} \cdot a^3}\). Understanding these properties can lead to simpler and more elegant solutions in algebraic problems.
When dealing with square roots, always consider the domain of the variables involved to ensure all expressions make sense in context.

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to their simplest form without changing their value. This process often includes applying laws of exponents, properties of square roots, and factoring techniques.
In our step-by-step solution, after multiplying and applying the laws of exponents, we simplified \(\sqrt{a^{(2n+5)+3}}\) to \(\sqrt{a^{2n+8}}\). Each step in the simplification process depends on understanding and applying mathematical rules accurately.

• Always ensure that you combine like terms when possible.
• Watch for opportunities to factor or expand expressions, which can often lead to a simpler form.

Mastering simplification techniques aids in solving equations efficiently and prepares you for more advanced algebraic concepts.