Problem 46 - Algebra 2

Question

Critical Thinking Describe the possible values of \(a\) such that \(\sqrt{72}+\sqrt{a}\) can be simplified to a single term.

Step-by-Step Solution

Answer

The possible value for \(a\) is 2.

Step 1: Factorize the number under the square root

To start off, we need to factorize 72. 72 can be written in several different ways, however, we are interested in integers that we can take the square root of. In the prime factorisation of 72 is \(2^3 * 3^2\), which contains two factors that are perfect squares, namely \(2^2\) and \(3^2\). Therefore we can write \(72 = 36 * 2\), where clearly the square root of \(36\) is an integer.

Step 2: Set up an equation

Now let's see if the square root of \(a\) can be a certain value such that the whole term simplifies to a single term. This can be achieved if the square root of \(a\) can be written as \(\sqrt{2}\). In order to get this value, \(a = 2\), since \(2\) is the number which, when square rooted, results in \(\sqrt{2}\). By equating the expressions, we get an equation: \(\sqrt{72}+\sqrt{2}= x\), where \(x\) represents the single term expression.

Step 3: Determine the possible value of a

Now, it remains to check if the equation is valued. If we substitute \(a = 2\) into the original equation, we get: \(\sqrt{72}+\sqrt{2}\) = \(\sqrt{36*2}+\sqrt{2}\) = \(6\sqrt{2}+\sqrt{2}\) = \(7\sqrt{2}\). This clearly allows the expression to be simplified to a single term, so we get that the only value of \(a = 2\).

Key Concepts

Prime Factorization

Prime factorization is a way of expressing a number as a product of prime numbers. It is a fundamental concept in mathematics, especially when dealing with roots and simplifying expressions.
Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
This means numbers like 2, 3, 5, 7, 11, and so forth.To factorize a number, you repeatedly divide by the smallest prime number until you are left with 1. For example, with the number 72:

• 72 is divided by 2, giving us 36.
• 36 is divided by 2, giving us 18.
• 18 is divided by 2, giving us 9.
• 9 is divided by 3, giving us 3.
• 3 is divided by 3, giving us 1.

Thus, the prime factorization of 72 is expressed as \(2^3 \times 3^2\).
This is useful because when dealing with square roots, identifying pairs of factors allows us to simplify the expression more easily.

Simplification of Expressions

Simplifying expressions, especially those involving square roots, makes calculations more manageable and easier to understand. The aim is to write the expression in its simplest form.
The expression \(\sqrt{72}\) is easier to work with if factorized into its component squares. As shown, the factors of 72 include perfect squares: \(36\), which is \(6^2\).
So, \(\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}\).Let's break it down further:

• Identify perfect square factors. Here, \(36 = 6^2\).
• Break the square root expression at these perfect square factors.
• Simplify the expression by taking the square root of perfect squares. For instance, \(\sqrt{36} = 6\).

This process not only reduces complexity but helps in adding, subtracting, and comparing root expressions.

Critical Thinking in Mathematics

Critical thinking in mathematics involves analyzing and evaluating an issue to formulate a clear and logical solution.
This process goes beyond simple calculations and encourages exploring different approaches and reasonings to arrive at a solution.When attempting to simplify \(\sqrt{72} + \sqrt{a}\) into a single term, critical thinking skills are employed to:

• Examine the problem: Why are certain values of \(a\) suitable?
• Utilize prime factorization to determine if the number under the square root has been optimized.
• Predict and test possible values for \(a\). Here, analyzing leads to identifying \(a = 2\) so that \(\sqrt{a} = \sqrt{2}\).

By applying these steps, you develop a deeper understanding of algebraic structures and enhance problem-solving skills, ensuring accuracy and efficiency in mathematical tasks.