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Chapter 16: Problem 21

Short Response Simplify \(\sqrt{3} \cdot \sqrt{18}\).

Short Answer

Expert verified
The simplified form is \(3 \sqrt{6}\).

Step by step solution

01

Simplify the Product Under a Single Radical

Since the product of square roots is the square root of the product, combine the square roots: \( \sqrt{3} \cdot \sqrt{18} = \sqrt{3 \times 18} \).
02

Calculate the Product Inside the Square Root

Multiply the numbers inside the square root: \( \sqrt{3 \times 18} = \sqrt{54} \).
03

Factor Inside the Radical

To simplify \( \sqrt{54} \), find the prime factorization of 54: \( 54 = 2 \times 3^3 \).
04

Simplify the Radical

Extract the square out of the radical: \( 3^2 = 9 \), which is a perfect square, so \( \sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \cdot \sqrt{6} = 3 \sqrt{6} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Roots
When we say square root, it's all about finding what number, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 times 4 equals 16.
Understanding square roots is crucial in simplifying expressions under radicals. Simplifying the expression \( \sqrt{3} \times \sqrt{18} \) involves multiplying the numbers and then simplifying the result.
Here鈥檚 a handy tip: the square root of a product is the same as the product of the square roots. This is why \( \sqrt{3} \times \sqrt{18} = \sqrt{54} \).
Whenever working with square roots:
  • Ensure you correctly multiply the values under the radicals.
  • Look for perfect squares within the result to simplify further.
Prime Factorization
Prime factorization is a method of expressing a number as a product of its prime numbers. For example, the number 54 can be broken down into 2 and the prime number 3, as \( 54 = 2 \times 3^3 \).
Identifying prime factors is a valuable skill in simplifying radicals because it helps to identify perfect squares within a number. Once the prime factors are determined, you can find pairs of factors that make perfect squares.
For the number 54:
  • It breaks down into \( 2 \times 3 \times 3 \times 3 \).
  • You can group the 3's: \( 3 \times 3 = 9 \), a perfect square.
Once you have the perfect squares, you can simplify the radical easily. Knowing factorization techniques makes working with radicals much easier.
Perfect Squares
Perfect squares are numbers that are the square of an integer. Numbers like 4, 9, 16, and 25 are perfect squares because they can be written as 2\( ^2 \), 3\( ^2 \), 4\( ^2 \), and 5\( ^2 \) respectively.
When simplifying radicals, identifying any factors of the number inside the radical that are perfect squares is key.
In the expression \( \sqrt{54} \), once you find that 9 is a factor, since \( 54 = 9 \times 6 \), you can use the fact that \( \sqrt{9} \) equals 3.
This lets you take the perfect square out of the radical, leaving: \( 3\sqrt{6} \).
Remember, working with perfect squares:
  • Helps in reducing the expression under the radical.
  • Makes the expression simpler and easier to understand.

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