Problem 15 - Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ \sqrt{3} \sqrt{15} $$

Step-by-Step Solution

Answer

The simplified form is \( 3\sqrt{5} \).

Step 1: Identify the Expression

Notice that the expression to be simplified is \( \sqrt{3} \cdot \sqrt{15} \). This involves multiplying two square roots together.

Step 2: Use the Product Property of Square Roots

Apply the property of square roots that states \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). Therefore, \( \sqrt{3} \cdot \sqrt{15} = \sqrt{3 \cdot 15} \).

Step 3: Multiply Inside the Square Root

Calculate the product inside the square root: \( 3 \cdot 15 = 45 \). So, \( \sqrt{3 \cdot 15} = \sqrt{45} \).

Step 4: Simplify the Square Root

Find the prime factorization of 45 to simplify: \( 45 = 3^2 \cdot 5 \). Use this to simplify the square root: \( \sqrt{45} = \sqrt{3^2 \cdot 5} = 3\sqrt{5} \).

Key Concepts

Product Property of Square Roots

Multiplying square roots can seem tricky at first, but with the product property of square roots, things become much simpler and more intuitive. This property allows you to break down the problem by stating that the product of two square roots is the same as the square root of their product. In mathematical form, this is expressed as:

• \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \)

For example, if you have \( \sqrt{3} \cdot \sqrt{15} \), you can apply this property to combine them into a single square root \( \sqrt{3 \cdot 15} \). Breaking down square roots in this way helps to streamline calculations and reduce complications when solving expressions that involve multiple square roots.

Simplifying Square Roots

Once you've combined square roots using the product property, it's time to simplify them further for a cleaner answer. The process involves rewriting the expression inside the square root as a product of numbers that include perfect squares, which makes it easier to "take out" these squares.

• For instance, in \( \sqrt{45} \), we can factor 45 into 9 and 5, since 9 is a perfect square.
• This allows us to rewrite it as \( \sqrt{9 \cdot 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5} \).

Breaking the number inside the square root down in this way allows you to extract perfect squares, thus simplifying the square root and making further mathematical manipulations more straightforward.

Prime Factorization

Prime factorization is a key technique in simplifying square roots, and understanding it can greatly enhance your mathematical toolkit.

• To find prime factors, you break down a number into its prime components, which are numbers that only have divisors of 1 and themselves.
• Take the number 45 as an example. By continually dividing by the smallest prime numbers, we find that 45 = 3\(^2\) \cdot 5.

Knowing the prime factorization helps us simplify square roots, because once numbers are expressed as a product of prime numbers and their powers, you can easily identify perfect squares. In \( \sqrt{45} \), recognizing that 9 is a perfect square, we extract it from under the root: \( \sqrt{3^2 \cdot 5} = 3\sqrt{5} \). This systematic breakdown transforms what seems like a complex problem into a manageable solution.