Question

Simplify each expression. \(\sqrt{45}\)

Short answer

The simplified form of \( \sqrt{45} \) is \( 3\sqrt{5} \).

Step-by-step solution

Identify Perfect Square Factor

To simplify \( \sqrt{45} \), we start by identifying the largest perfect square factor of 45. The factors of 45 are 1, 3, 5, 9, 15, 45. Among these, 9 is a perfect square (\( 3^2 \)).

Rewrite the Radicand

Express 45 in terms of its perfect square factor: \( 45 = 9 \times 5 \). Thus, we can rewrite the expression under the square root as \( \sqrt{9 \times 5} \).

Apply the Product Rule of Square Roots

According to the product rule, \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). Applying this rule, we have \( \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} \).

Simplify Each Square Root

The square root of 9 is 3, so we have \( \sqrt{9} = 3 \). Therefore, \( \sqrt{9} \times \sqrt{5} = 3 \times \sqrt{5} \).

Complete the Simplification

Finally, combine the simplified terms to get \( 3\sqrt{5} \). This is the simplified form of \( \sqrt{45} \).

Key concepts

Perfect Square Factor

When simplifying square roots, the perfect square factor is a key component. A perfect square is a number that can be expressed as another whole number squared, such as 1, 4, 9, 16, etc. - To simplify a square root, it's essential to identify the largest perfect square factor of the number under the square root, also known as the radicand. - For example, in the simplification of \( \sqrt{45} \), we break down the number 45 into its factors: 1, 3, 5, 9, 15, and 45. - Among these, 9 is a perfect square since it equals \( 3^2 \).By identifying 9 as a factor of 45, we can simplify the square root by leveraging this perfect square, making our calculations easier and resulting in a neat simplification.

Product Rule of Square Roots

The product rule of square roots simplifies the process of separating or combining the roots of two or more numbers. This rule states that the square root of a product is equal to the product of the square roots of each factor: \[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \].- In the context of simplifying \( \sqrt{45} \), once we express 45 as 9 times 5, we apply the product rule. - This allows us to rewrite \( \sqrt{9 \times 5} \) as \( \sqrt{9} \times \sqrt{5} \).By applying this principle, we isolate the perfect square (9), and simplify it, while leaving non-perfect square components (5) intact under the square root.

Radicand

The radicand is the number located underneath the symbol of the square root. This number is crucial because it determines what we are attempting to simplify or square root. - In our example, \( \sqrt{45} \), the radicand is 45. - Our task is to simplify this radicand by utilizing the factors it comprises.By rewriting the radicand in terms of its perfect square factor, as 9 multiplied by 5, we set up the equation to efficiently apply the product rule of square roots.Understanding and manipulating the radicand is one of the first steps in simplifying any square root and is foundational to mastering square root simplifications.