Write $12 \sqrt{8}$ in simple radical form.
Answer (2)
Factor the radicand: 8 = 2 2 × 2 .
Rewrite the expression: 12 8 = 12 2 2 × 2 .
Separate the square root: 12 2 2 × 2 = 12 × 2 2 × 2 = 12 × 2 × 2 .
Simplify: 12 × 2 × 2 = 24 2 . The final answer is 24 2 .
Explanation
Understanding the Problem We want to simplify the expression 12 8 into its simplest radical form. This means we want to write the expression such that the number under the square root (the radicand) has no perfect square factors other than 1.
Factoring the Radicand First, let's factor the number under the square root, which is 8. We can write 8 as a product of its prime factors: 8 = 2 × 2 × 2 = 2 3 .
Rewriting the Expression Now we can rewrite the original expression as 12 2 3 . Since 2 3 = 2 2 × 2 , we have 12 2 2 × 2 .
Separating the Square Root Using the property of square roots, a × b = a × b , we can separate the square root: 12 2 2 × 2 = 12 × 2 2 × 2 .
Simplifying the Square Root Since 2 2 = 2 , we can simplify the expression further: 12 × 2 × 2 .
Multiplying the Constants Finally, we multiply the constants: 12 × 2 = 24 , so the simplified expression is 24 2 .
Final Answer Therefore, 12 8 in simple radical form is 24 2 .
Examples
Understanding how to simplify radicals is useful in many areas, such as physics and engineering, where you might need to calculate distances or areas. For example, if you are calculating the length of the diagonal of a square with side length 2 , you would use the Pythagorean theorem: d 2 = ( 2 ) 2 + ( 2 ) 2 = 2 + 2 = 4 . Then, d = 4 = 2 . Simplifying radicals helps in obtaining exact values and making calculations easier.
The expression 12 8 simplifies to 24 2 by factoring 8, separating the square root, and performing simple multiplication. This method ensures all radical forms are in their simplest state. Understanding this process is crucial for mastering algebraic simplifications.
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