Question

Simplify each expression. \(\sqrt{32}\)

Short answer

\(\sqrt{32} = 4\sqrt{2}\)

Step-by-step solution

Identify Perfect Squares

To simplify the square root of 32, the first step is to identify any perfect square factors of 32. Recall that a perfect square is a number that can be expressed as the square of an integer. In the case of 32, we can break it down into factors: 32 = 16 x 2. Here, 16 is a perfect square since 16 = 4^2.

Rewrite the Radical

Use the factors identified in Step 1 to rewrite the square root of 32. You can express it as the product of the square roots of those factors:\(\sqrt{32} = \sqrt{16 \times 2}\)

Apply the Product Property of Square Roots

The product property of square roots states that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). Using this property, we can separate the square root of the product into the product of the square roots:\(\sqrt{32} = \sqrt{16} \times \sqrt{2}\)

Simplify the Expression

Simplify \(\sqrt{16}\) since 16 is a perfect square:\(\sqrt{16} = 4\)Thus, the expression becomes:\(4 \times \sqrt{2}\)

Write the Final Answer

Combine all parts from the previous steps to write your final simplified expression:\(\sqrt{32} = 4 \sqrt{2}\)

Key concepts

Perfect Squares

Perfect squares play a crucial role when simplifying radicals. A perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it equals the square of 4, written as
\( 16 = 4^2 \).
This concept is vital because recognizing perfect squares helps break down larger numbers into simpler components.
When tackling the square root of a number, such as 32, it is helpful to find any perfect square factors. Here, 32 can be expressed as the product
\( 32 = 16 \times 2 \).
Identifying 16 as a perfect square is the first step in simplification, which helps us rewrite and simplify the expression more efficiently.

Product Property of Square Roots

The product property of square roots is a useful tool that helps simplify expressions involving square roots. This property states that the square root of a product is equal to the product of the square roots of each factor:
\( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).
This property allows us to split a more complex square root into manageable parts.
For example, in the expression \( \sqrt{32} \), using the product property of square roots helps us rewrite it as
\(\sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} \).
By doing this, we isolate the perfect square (16), simplifying the calculation. The product property is significant because it makes working with radicals more straightforward and less intimidating.

Step-by-Step Solutions

Breaking down a math problem into step-by-step solutions is a strategy that simplifies the learning process. It involves slowly moving through each component of a problem and addressing them individually. This approach makes complex problems easier to understand.
For simplifying radicals, as in \( \sqrt{32} \), our step-by-step solution begins with identifying perfect squares:

• Step 1: Identify factors, recognizing 16 as a perfect square.
• Step 2: Rewrite the expression using the factors: \( \sqrt{16 \times 2} \).
• Step 3: Apply the product property: \( \sqrt{16} \times \sqrt{2} \).
• Step 4: Further simplification by evaluating \( \sqrt{16} = 4 \).
• Step 5: Combine all parts to reach the final answer: \( 4 \sqrt{2} \).

Using a step-by-step approach helps in grasping the reasoning behind
each transformation, building confidence in solving radical expressions.