Problem 67 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{2 m^{3 n+1}} \sqrt{10 m^{n+3}} $$

Step-by-Step Solution

Answer

Answer: The product of the given square roots is $2 m^{2(n+1)}\sqrt{5}$.

Step 1: Write down the given expression

The given expression is: $$ \sqrt{2 m^{3 n+1}} \sqrt{10 m^{n+3}} $$

Step 2: Combine the square roots

Using the property of square roots for products, we get: $$ \sqrt{2 m^{3 n+1}} \sqrt{10 m^{n+3}} = \sqrt{(2 m^{3 n+1})(10 m^{n+3})} $$

Step 3: Simplify the expression inside the square root

Now, we will simplify the expression inside the square root. We can factor and use the addition rule of exponents: $$ \sqrt{(2 m^{3 n+1})(10 m^{n+3})} = \sqrt{20 m^{(3 n+1)+(n+3)}} $$

Step 4: Simplify the exponent

Next, we need to simplify the exponent of the expression: $$ \sqrt{20 m^{(3 n+1)+(n+3)}} = \sqrt{20 m^{4 n+4}} $$

Step 5: Factor the expression inside the square root

Lastly, we can factor the expression inside the square root as follows: $$ \sqrt{20 m^{4 n+4}} = \sqrt{4 \cdot 5m^{4(n+1)}} $$

Step 6: Simplify the square root

Finally, we simplify the square root and obtain the result: $$ \sqrt{4 \cdot 5m^{4(n+1)}} = 2 m^{2(n+1)}\sqrt{5} $$ So, the product of the given square roots is: $$ 2 m^{2(n+1)}\sqrt{5} $$

Key Concepts

Square Roots

Understanding square roots is crucial in algebra. A square root of a number is a value that, when multiplied by itself, gives the original number. It's symbolized by the radical sign \( \sqrt{} \). For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \). Square roots have specific properties that make them powerful:

• The square root of a product is the product of the square roots: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).
• You can also find square roots of variables: if \( x^2 = a \), then \( x = \sqrt{a} \).

In expressions, handling large numbers or variables inside square roots can be simplified using these properties.

Exponent Rules

Exponent rules, also known as laws of exponents, are helpful for simplifying expressions involving powers. When numbers or variables are multiplied, and they have exponents, certain rules apply:

• Product of Powers Rule: \( x^a \times x^b = x^{a+b} \). This rule allows you to add exponents when you multiply like bases.
• Power of a Power Rule: \((x^a)^b = x^{a\times b}\). This rule is used when raising an exponent to another power.
• Power of a Product Rule: \((xy)^a = x^a \times y^a\). Each base in a product can be raised to the power separately.

These rules are foundational in simplifying algebraic expressions with multiple terms. They allow for combining and reducing expressions effectively.

Product of Radicals

The concept of the product of radicals comes into play when dealing with expressions under a radical (square root). By using the property of square roots that the square root of a product is equal to the product of the square roots, we can simplify expressions:

• \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \). This allows us to combine two square roots into one.

Taking the square root of a product helps in simplifying complex expressions. In the exercise, \( \sqrt{2 m^{3n+1}} \times \sqrt{10 m^{n+3}} = \sqrt{20 m^{3n+1+n+3}} \). This step consolidates two separate square roots into a single expression, paving the way for further simplification.

Simplification of Expressions

Simplification of expressions is the process of breaking down complex algebraic expressions into simpler forms. This often involves the use of exponent rules and square root properties:

• Combine like terms or factors, especially those that are under a square root, to ease simplification.
• Factor expressions under a square root to identify perfect squares, which can be simplified easily.

In practice, expressions like \( \sqrt{4 \cdot 5m^{4(n+1)}} \) can be broken down to \( 2 m^{2(n+1)}\sqrt{5} \) as seen in the exercise. This means taking the square root of perfect squares (like 4) outside the radical and reducing the complexity of the expression. Simplification makes algebraic manipulation clearer and results more accessible.