Problem 60 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{7(2 k-1)^{11}(k+1)^{3}} \sqrt{14(2 k-1)^{10}} $$

Step-by-Step Solution

Answer

Answer: The product of the two square roots is $\sqrt{98(2k-1)^{21}(k+1)^{3}}$.

Step 1: Simplify the expressions inside the square roots.

In order to simplify the expressions inside the square roots, we first need to identify any shared factors that can be simplified. We can note that there is a common factor between the expressions inside the square roots: (2k-1). It appears with different exponents in both expressions, so we can take the smallest exponent and simplify the expressions accordingly. $$ \sqrt{7(2 k-1)^{11}(k+1)^{3}} \sqrt{14(2 k-1)^{10}} = \sqrt{7(2 k-1)^{11}(k+1)^{3}\cdot14(2 k-1)^{10}} $$

Step 2: Combine the expressions inside the single square root and simplify.

Now, we can combine the expressions inside the single square root by multiplying them together and simplify the resulting expression. $$ \begin{align*} \sqrt{7(2 k-1)^{11}(k+1)^{3}\cdot14(2 k-1)^{10}} &= \sqrt{7\cdot14(2k-1)^{11}(2k-1)^{10}(k+1)^{3}} \\ &= \sqrt{98(2k-1)^{21}(k+1)^{3}} \end{align*} $$

Step 3: Write the final answer.

The simplified product of the two square roots is: $$ \sqrt{98(2k-1)^{21}(k+1)^{3}} $$

Key Concepts

Square Roots

Square roots are fundamental in math, helping us find the original number that was squared to get a specific result. When dealing with square root expressions, you're aiming to simplify them as much as possible. This often involves looking for squares inside the expression that can be easily extracted.

For example, consider \(\sqrt{a^2}\); the solution simplifies to \(a\) because \(a\) squared gives \(a^2\).

When you multiply two square roots, such as \(\sqrt{a} \times \sqrt{b}\), you can combine them under a single square root: \(\sqrt{ab}\). This approach is particularly useful when simplifying expressions that involve products under square roots, just like in our problem where we handle the square root of a product like \(\sqrt{98(2k-1)^{21}(k+1)^{3}}\). The key steps are:

• Combine the expressions under one square root when possible.
• Identify and simplify square factors wherever you find them.
• Simplify carefully to ensure you're following mathematical rules for square roots.

Exponents

Exponents represent how many times a number, known as the base, is multiplied by itself. When simplifying algebraic expressions, especially those involving exponents, it's important to remember the basic rules.

Consider the expression \((a^m \times a^n)\), which simplifies to \(a^{m+n}\). This rule is pivotal when combining like terms under square roots, such as \((2k-1)^{11} \cdot (2k-1)^{10}\), simplifying it to \((2k-1)^{21}\).

Here’s a quick checklist for working with exponents:

• Add exponents when multiplying like bases.
• Subtract exponents when dividing like bases.
• Multiply exponents when raising a power to another power, like \((a^m)^n = a^{mn}\).

Understanding and applying these rules is essential to efficiently simplify expressions involving exponential terms.

Algebraic Simplification

Algebraic simplification is the process of making an expression easier to understand or use. This involves reducing expressions to their simplest form by following the rules of algebra and arithmetic.

The goal is to shorten and simplify without changing the value. In the context of a problem, it often means recognizing common factors in expressions and simplifying them. For example, in \(\sqrt{7(2k-1)^{11}(k+1)^{3} \times 14(2k-1)^{10}}\), notice how both parts inside the root share the factor \((2k-1)\). This makes it possible to combine them as \((2k-1)^{21}\), greatly simplifying the expression.

Steps for simplification include:

• Identifying and factoring out common terms.
• Simplifying exponents by using the rules of exponents.
• Combining like terms whenever possible.

By focusing on these strategies, you can streamline complex expressions into more manageable forms.