Problem 77 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{12 m^{3}}\left(\sqrt{6 m^{7}}-\sqrt{3 m}\right) $$

Step-by-Step Solution

Answer

Answer: The simplified product of the given expression is $$\sqrt{72 m^{10}} - \sqrt{36 m^4}$$.

Step 1: Distribute the first term

Distribute the first term, $$\sqrt{12 m^3}$$, to both terms inside the parentheses: $$\sqrt{6 m^7}$$ and $$-\sqrt{3 m}$$. This means we have to take the product of the first term with each term separately.

Step 2: Simplify the expressions by multiplying

Take the product of each multiplied term and simplify: $$ \sqrt{12 m^3} \cdot \sqrt{6 m^7} = \sqrt{(12 m^3)(6 m^7)} = \sqrt{72 m^{10}} $$ And, $$ \sqrt{12 m^3} \cdot -\sqrt{3 m} = -\sqrt{(12 m^3)(3 m)} = -\sqrt{36 m^4} $$

Step 3: Combine the expressions

Now that we have simplified both the products, combine the resulting expressions to get the final answer: $$ \sqrt{72 m^{10}} - \sqrt{36 m^4} $$ Hence, the product of the given expression is: $$ \sqrt{72 m^{10}} - \sqrt{36 m^4} $$

Key Concepts

Simplifying Radicals

Simplifying radicals is a process in mathematics that involves making the expression under the square root as simple as possible. This means breaking it down to its prime factors and finding perfect squares. For instance, if we have \( \sqrt{72} \), it involves the following steps:

• Identify the prime factors of 72: That would be 2, 2, 2, 3, and 3.
• Group the factors into pairs: (2 \( \times \) 2) and (3 \( \times \) 3) with a single 2 left alone.
• The pair of 2 multiplies to give 4, a pair of 3 multiplies to give 9. The leftover 2 stays under the square root sign.
• This simplifies to \( \sqrt{4 \times 9 \times 2} = \sqrt{36 \times 2} = 6 \sqrt{2} \).

For variables, like \( m^{10} \), simplify by halving the exponent:

• The exponent 10 divides evenly by 2, resulting in \( m^5 \) coming out of the radical.
• Thus, \( \sqrt{m^{10}} = m^5 \).

Applying these methods helps in dealing with radical expressions easily and efficiently.

Distributive Property

The distributive property is a key principle in algebra where a single term is multiplied by terms inside a parenthesis. It's generally expressed as \( a(b + c) = ab + ac \). This principle distributes the multiplication over addition or subtraction inside the bracket. In the context of the given problem, your task is to apply this property to the radicals:

• Take \( \sqrt{12m^3} \) and multiply it through each term inside the parentheses, \( \sqrt{6 m^7} \) and \( -\sqrt{3 m} \).
• First simplifies to \( \sqrt{12m^3} \cdot \sqrt{6m^7} = \sqrt{72m^{10}} \).
• Second simplifies to \( \sqrt{12m^3} \cdot -\sqrt{3m} = -\sqrt{36m^4} \).

Using the distributive property correctly ensures every term inside the parentheses gets multiplied by the expression outside. It is specific, clear, and crucial to simplifying radical expressions correctly.

Algebraic Multiplication

Algebraic multiplication involves combining terms correctly according to algebra rules. When dealing with radicals, you multiply the numbers under the square roots separately and the variable parts separately. Here's a simple breakdown:

• When multiplying \( \sqrt{a} \) and \( \sqrt{b} \), such as \( \sqrt{12m^3} \times \sqrt{6m^7} \), you multiply them like normal numbers to get \( \sqrt{72m^{10}} \).
• This works because of the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \).
• Remember to follow the rules for exponents when variables are involved. Here, sum the exponents in \( m^3 \) and \( m^7 \) to produce \( m^{10} \).

Understanding these rules lets you handle not just numbers, but the variables too, ensuring you apply the multiplication correctly to simplify radical expressions. Be careful! Always check the signs, such as negative signs, during your multiplication to avoid errors.