What is 8 Square Root of 3 Squared?

Exponents are fundamental mathematical tools that allow us to represent repeated multiplication in a concise and powerful way. They appear in various forms, including whole number exponents, fractions, and even radicals. In this blog post, we’ll dive deep into the concept of exponents, specifically focusing on what happens when we square a quantity that contains both an exponent and a radical. We’ll explore the intricacies of (8√3)^2, unraveling its meaning and significance.

What is 8 Square Root of 3 Squared?

The value of 8 times the square root of 3 squared is 24. When you square the square root of 3 (which is 3) and then multiply it by 8, you get the result of 24.

The Basics: Exponents and Radicals

Before we delve into (8√3)^2, let’s review some key concepts.

• Exponents: An exponent tells us how many times a base number should be multiplied by itself. For instance, 2^3 means 2 multiplied by itself three times, resulting in 2 * 2 * 2 = 8.
• Radicals: Radicals are expressions that include square roots (√). The square root of a number, denoted as √x, is the value that, when multiplied by itself, equals x. For example, √4 is 2 because 2 * 2 = 4.

(8√3)^2: Breaking It Down

Now, let’s tackle (8√3)^2 step by step.

1. Squaring the Quantity: To square (8√3), we multiply it by itself:(8√3)^2 = (8√3) * (8√3)
2. Using the Distributive Property: We can apply the distributive property here:(8√3) * (8√3) = 8 * 8 * (√3 * √3)
3. Simplifying the Multiplication:
   • 8 * 8 = 64
   • (√3 * √3) = √(3 * 3) = √9 = 3
4. Putting It All Together:(8√3)^2 = 64 * 3 = 192

Conclusion: (8√3)^2 Equals 192

The result of (8√3)^2 is 192. This means that when you square the quantity 8√3, you obtain 192.

Interpreting the Result

What does this result signify? In essence, it tells us that when you square a number that contains both an exponent and a radical, you apply the exponent to the number itself and the exponent inside the radical separately. In this case, 8^2 is 64, and (√3)^2 is 3, and when multiplied together, they yield 192.

Applications of Exponents and Radicals

Understanding how exponents and radicals interact is crucial in various fields, such as science, engineering, and finance. These concepts enable us to handle complex calculations efficiently, solve equations, and model real-world phenomena.

In conclusion, (8√3)^2 simplifies to 192, demonstrating the synergy between exponents and radicals. The beauty of mathematics lies in its ability to unveil hidden patterns and simplify seemingly complex expressions, making it an indispensable tool in our quest for knowledge and understanding.