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root if 128 is what number

root if 128 is what number

2 min read 21-01-2025
root if 128 is what number

The question, "What is the square root of 128?" might seem simple at first glance. However, understanding how to solve this and expressing the answer correctly involves several key mathematical concepts. This guide will break down the process step-by-step, exploring both the approximate decimal value and the simplified radical form.

Understanding Square Roots

Before we dive into the specifics of the square root of 128, let's review the fundamental concept. The square root of a number is a value that, when multiplied by itself (squared), equals the original number. For example, the square root of 9 is 3 because 3 x 3 = 9.

Calculating the Square Root of 128

128 isn't a perfect square (a number that results from squaring a whole number). Therefore, we won't get a whole number as our answer. There are two ways to approach this:

1. Approximating the Decimal Value

Using a calculator, we find that the square root of 128 is approximately 11.3137. This is a decimal approximation, accurate to several decimal places.

2. Simplifying the Radical

We can also express the square root of 128 in its simplest radical form. This involves finding perfect square factors within 128.

  • Factor 128: 128 can be factored as 64 x 2.
  • Identify Perfect Squares: Notice that 64 is a perfect square (8 x 8 = 64).
  • Simplify: We can rewrite √128 as √(64 x 2). Because the square root of a product is the product of the square roots, this simplifies to √64 x √2.
  • Final Answer: Since √64 = 8, the simplified radical form is 8√2.

This means 8√2 is the exact value of the square root of 128. The decimal approximation (11.3137) is just a close numerical representation.

Why Both Forms Are Important

Both the decimal approximation and the simplified radical form are valuable:

  • Decimal Approximation: Useful for practical calculations and applications where a precise numerical value is needed.
  • Simplified Radical Form: Provides the exact mathematical representation and is often preferred in algebra and other mathematical contexts. It avoids rounding errors inherent in decimal approximations.

Frequently Asked Questions (FAQs)

Q: How do I calculate square roots without a calculator?

A: For perfect squares, you can use memorization or simple multiplication. For non-perfect squares like 128, you'll need methods like the Babylonian method (a type of iterative algorithm) or long division. However, calculators are generally the most efficient tool for these calculations.

Q: What are some real-world applications of square roots?

A: Square roots appear in numerous fields, including:

  • Physics: Calculating distances, speeds, and accelerations.
  • Engineering: Designing structures, analyzing forces, and solving geometric problems.
  • Finance: Determining investment returns and calculating compound interest.
  • Geometry: Finding lengths of diagonals in squares and rectangles.

Conclusion

The square root of 128 is approximately 11.3137 (decimal form) or exactly 8√2 (simplified radical form). Understanding both representations is crucial for various mathematical applications. Remember that the simplified radical form offers precision, while the decimal approximation is practical for everyday use. Now you have a solid understanding of how to solve this type of problem and why both answers are important!

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