Problem 57 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{12(p-q)^{3}} \sqrt{3(p-q)^{5}} $$

Step-by-Step Solution

Answer

Answer: The product of the given expressions is \(\sqrt{36(p-q)^{8}}\).

Step 1: Identify the given expressions

We have the following expressions to multiply: $$ \sqrt{12(p-q)^{3}} \quad \text{and} \quad \sqrt{3(p-q)^{5}} $$

Step 2: Apply the rule for multiplying square roots

We will apply the rule \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\) to combine the two expressions: $$ \sqrt{12(p-q)^{3}} \cdot \sqrt{3(p-q)^{5}} = \sqrt{(12(p-q)^{3})(3(p-q)^{5})} $$

Step 3: Simplify the expression inside the square root

Now, we need to simplify the expression inside the square root: $$ \sqrt{(12(p-q)^{3})(3(p-q)^{5})} = \sqrt{36(p-q)^{8}} $$

Step 4: Final answer

Therefore, the product of the given expressions is: $$ \sqrt{12(p-q)^{3}} \sqrt{3(p-q)^{5}} = \sqrt{36(p-q)^{8}} $$

Key Concepts

Simplifying Square Roots

Understanding how to simplify square roots is essential in solving many algebra problems efficiently. A square root, represented by the radical symbol \( \sqrt{} \), asks for a number which, when multiplied by itself, gives the original number inside the radical. To simplify a square root, the goal is to find the prime factorization of the number inside the radical and then pair identical factors. For instance, to simplify \( \sqrt{12} \), we break down 12 into its prime factors:\[ 12 = 2 \times 2 \times 3 \]

Now, we can pair the twos to remove them from under the radical: \[ \sqrt{12} = \sqrt{2 \times 2 \times 3} = 2 \sqrt{3} \].

This process also applies when variables with exponents are inside the square root—the exponents can also be simplified using the same pairing technique. In the exercise provided, \( \sqrt{12(p-q)^3} \sqrt{3(p-q)^5} \) involves simplifying by pairing factors within the radicand (the number inside the radical).

Exponent Rules

When dealing with radical expressions, we often encounter exponents. It is crucial to remember some basic exponent rules to simplify these expressions effectively. One of the key rules is that when you multiply two powers with the same base, you can add the exponents. In mathematical terms, \( a^m \cdot a^n = a^{m+n} \).

Using this rule in the given exercise, we can combine the exponents of \( (p-q) \) as follows: \[ (p-q)^3 \cdot (p-q)^5 = (p-q)^{3+5} = (p-q)^8 \].

By applying these exponent rules, multiplication of radical expressions becomes significantly more manageable. Additionally, if an exponent is even and we're working under a square root, such as \( (p-q)^8 \), we can take half of that exponent out of the radical, which simplifies the expression further.

Radical Expressions

Radical expressions involve roots—such as square roots, cube roots, and other higher-order roots. Multiplying radical expressions with the same index (which is 2 for square roots) follows a straightforward rule: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). This simple but powerful rule allows us to combine and simplify expressions under a single radical, as in the example from the textbook exercise.

It's also important to understand that radical expressions can include not only numerical values but also algebraic expressions. When multiplying radicals that contain variables, as in the exercise \( \sqrt{12(p-q)^3} \cdot \sqrt{3(p-q)^5} \) presented here, we apply the same principles as we do with numbers—combine the radicands first and then simplify wherever possible. Additionally, recognizing perfect squares under the radical will help in further simplifying the expression. For example, \( \sqrt{36(p-q)^8} \) simplifies easily since 36 is a perfect square and \( (p-q)^8 \) can be divided by 2, the index of the square root.