Problem 7 - Elementary and Intermediate Algebra

Question

Multiply. $$ \sqrt{5} \sqrt{7} $$

Step-by-Step Solution

Answer

\(\sqrt{5} \sqrt{7} = \sqrt{35}\)

Step 1: Understand the Multiplication of Radicals

When multiplying two square roots, the product is the square root of the product of the radicands (the numbers inside the square roots).

Step 2: Multiply the Radicands

Multiply the numbers inside the square roots: \(5 \times 7 = 35\).

Step 3: Take the Square Root of the Product

The result of multiplying \( \sqrt{5} \times \sqrt{7} \) is \sqrt{35}\.

Key Concepts

square roots

Square roots are one of the basic operations in mathematics. A square root removes the square from a number. For example, the square root of 9 is 3, because 3 squared (3×3) equals 9.
Square roots are commonly denoted by the radical symbol (√). So, \(\sqrt{16} = 4\).
That is because 4 times 4 equals 16.
Remember: A square root takes us back to the number that was squared.

radicands

The radicand is the number inside the square root symbol. When we write \(\sqrt{25}\), 25 is the radicand.
This value is what we find the square root of.
In our exercise, \(\sqrt{5} \text{ and } \sqrt{7}\), the numbers 5 and 7 are the radicands.
During multiplication, we multiply these radicands together like normal numbers. So, in our example, 5 multiplied by 7 gives us 35.
This means that our product of radicals will have 35 as its new radicand.

product of radicals

To multiply radicals, we must follow some easy steps:

• First, multiply the radicands (numbers inside the square roots).
• Next, place the product inside a new square root symbol.

Let's apply these steps: In our given problem, we have \(\sqrt{5} \times \sqrt{7}.\)
Following the steps, we multiply 5 and 7, getting 35.
Then we place this product inside a square root, resulting in \(\sqrt{35}.\)
This product of radicals simplifies the expression and shows us the answer.