Simplifying Radicals: A Step-by-Step Guide to Simplifying -52 5√40

Dealing with complex algebraic expressions can be challenging, especially when it comes to simplifying radicals. In this article, we will guide you through the process of simplifying the expression -52 5√40, demonstrating both the step-by-step approach and the final answer. This will help you understand the underlying principles and improve your problem-solving skills in algebra.

Introduction to Radicals

Radicals, or square roots, are a fundamental part of algebra. The square root of a number, denoted as √, is a value that, when multiplied by itself, gives the original number. For instance, √16 4 because 4 × 4 16.

Step-by-Step Simplification of -52 5√40

Let's start with the expression: -52 5√40.

Step 1: Factorize the Radicand

The first step in simplifying a radical is to factorize the radicand (the number under the square root) into its prime factors. In this case, 40 can be factorized as follows:

40 2 x 20 20 2 x 10 10 2 x 5

So, 40 2 x 2 x 2 x 5.

Step 2: Rewrite the Expression

Now, let's rewrite the expression using the prime factors of 40:

-52 5√40 -52 5√(2 x 2 x 2 x 5).

Step 3: Group the Perfect Squares

In the expression -52 5√(2 x 2 x 2 x 5), we can see that 2 x 2 is a perfect square. We can take the square root of 2 x 2, which is 2, and bring it outside the square root.

-52 5√(2 x 2 x 2 x 5) -52 5√((2 x 2) x 2 x 5) -52 5√(4 x 2 x 5).

Step 4: Simplify Further

Now, we can simplify the expression further by taking the square root of the perfect square (4) and multiplying it by the coefficient outside the radical:

-52 5√4 x 2 x 5 -52 5 x 2 √(2 x 5) -104 √(10).

Step 5: Convert to Decimal Form

Finally, let's convert the simplified expression -104√10 to its decimal form. Using a calculator, we get:

-104 √10 ≈ -327.067.

Conclusion

In conclusion, we simplified the expression -52 5√40 to -104√10, which in decimal form is approximately -327.067. Mastering the steps outlined above will help you simplify similar radical expressions in the future. Understanding the process not only improves your algebraic skills but also enhances your problem-solving abilities.

Keywords: simplifying radicals, square roots, algebraic expressions.

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