Problem 6 - Algebra for College Students

Question

Multiply and simplify where possible. \((-7 \sqrt{3})(2 \sqrt{5})\)

Step-by-Step Solution

Answer

The result is \(-14 \sqrt{15}\).

Step 1: Multiply the Coefficients

To begin, we look at the coefficients of the two expressions, which are -7 and 2. When we multiply these coefficients together, we get \(-7 \times 2 = -14\).

Step 2: Multiply the Radicals

Next, we need to multiply the square root parts. The radicals are \( \sqrt{3} \) and \( \sqrt{5} \). By multiplying these radicals, we obtain \( \sqrt{3} \times \sqrt{5} = \sqrt{15} \).

Step 3: Combine the Results

Now, we combine the results from Step 1 and Step 2. The product of the coefficients \(-14\) is multiplied by the product of the radicals\( \sqrt{15} \). This gives us the final expression:\(-14 \sqrt{15}\).

Key Concepts

Coefficients

Coefficients are the numerical factors in terms of an algebraic expression. In radical expressions like \((-7 \sqrt{3})(2 \sqrt{5})\), coefficients are the numbers located before the square root signs, which in this case are -7 and 2. Understanding coefficients is crucial because they tell us how many times a radical is being multiplied. To multiply the coefficients, you simply multiply the numbers together.

• Example: Multiply -7 and 2, resulting in -14.

This multiplication is straightforward and follows the same rules you would use when multiplying any integers. Coefficients work together with radicals to form simplified expressions where both parts must be considered individually.
Make sure you treat coefficients and radicals separately when you're beginning calculations. Only once they are both worked out individually, should you combine the results, as seen in the solution step where the product of coefficients was -14.

Simplifying Expressions

Simplifying expressions involves making a mathematical expression as basic and clear as possible without changing its value. When working with radical expressions, simplifying can include combining like terms, reducing fractions, or combining radical expressions.
In radical expressions like \(-14 \sqrt{15}\), checking for simplification usually involves looking at the square root part.

• First, check if the number under the square root (the radicand) can be simplified further.
• In our example, \(\sqrt{15}\) is already in its simplest form as 15 is the product of 3 and 5, both prime numbers.

Sometimes, simplifying radicals can involve breaking down the radicand into factors and simplifying the expression further this way.
However, if you cannot simplify the radical part further, then the expression is fully simplified as it is. In our example, -14 remains as the coefficient and \(\sqrt{15}\) stays as it is, making \(-14 \sqrt{15}\) the simplest form of the expression.

Square Roots

Square roots, often represented with the radical symbol \(\sqrt{}\), are used to find a number which, when multiplied by itself, gives the original number under the root.
For instance, \(\sqrt{4} = 2\), because when 2 is multiplied by itself, it results in 4.
When dealing with square roots, you may also encounter expressions that include radicals like \(\sqrt{3}\) or \(\sqrt{5}\). In these cases, multiplying square roots involves using the product property of square roots:

• \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\)

Using this property, you simplify expressions such as the one given: multiplying \(\sqrt{3}\) by \(\sqrt{5}\) results in \(\sqrt{15}\). This way, square roots can be combined to form a new simpler expression, as long as the radicands are positive numbers.
The product property is extremely useful for simplifying and performing operations with radicals efficiently.