Question

Simplify each expression. \(\sqrt{5} \cdot \sqrt{15}\)

Short answer

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

Step-by-step solution

Use the Property of Square Roots

We begin by using the property that states \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). Apply this property to simplify \(\sqrt{5} \cdot \sqrt{15}\).

Multiply Inside the Square Root

Next, multiply the numbers inside the square root: \(5 \times 15 = 75\). This gives us \(\sqrt{75}\).

Factor the Expression Inside the Square Root

Write 75 as a product of its prime factors: \(75 = 3 \times 5^2\).

Simplify the Square Root

Since \(\sqrt{5^2} = 5\), extract 5 from the square root: \(\sqrt{3 \times 5^2} = 5\sqrt{3}\). This simplifies \(\sqrt{75}\) to \(5\sqrt{3}\).

Key concepts

Properties of Square Roots

The properties of square roots help us simplify expressions that involve radicals. One important property is the multiplication property, which states that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). This allows us to combine two square roots into one. Following this property, multiplying square roots is as simple as multiplying the numbers inside the radicals and then simplifying if possible.

• This is useful for expressions where you can't easily take the square root of an individual number but can when combined with another number.
• It also makes complex calculations easier by reducing the number of steps required.

Understanding this property allows us to effectively tackle problems like \( \sqrt{5} \cdot \sqrt{15} \), turning it into \( \sqrt{75} \) quickly, setting us up for further simplification.

Multiplying Square Roots

When multiplying square roots, the first step is to use the property you already learned: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). Start by multiplying the numbers inside the square roots. In our example, \( \sqrt{5} \cdot \sqrt{15} \), you calculate \( 5 \times 15 \), which equals 75.
Here’s a quick guide:

• Always multiply the numbers inside first.
• If the product is a perfect square, simplify it directly.
• If not, proceed to factorize and simplify as needed.

By keeping these steps in mind, you can streamline your approach to multiplying square roots and handle a range of problems with ease.

Prime Factorization

Prime factorization is the process of breaking down a number into its basic prime components. For example, the number 75 can be expressed as \( 3 \times 5^2 \). This is useful when simplifying square roots because it helps identify which numbers can be "pulled out" of the root.
Steps for prime factorization:

• Divide the number by the smallest prime number (e.g., 2, 3, 5, 7) until it is no longer divisible.
• Repeat with the next smallest prime number.
• Continue until all factors are prime.

In the example \( \sqrt{75} \), recognizing that 75 is \( 3 \times 5^2 \) allows us to remove \( 5^2 \) from under the square root as a single 5, simplifying the expression to \( 5\sqrt{3} \). Knowing how to find prime factors makes it much easier to simplify square roots effectively.