What Is The Square Root Of 84

Unveiling the Mystery: Finding the Square Root of 84

Understanding square roots is fundamental to various mathematical fields, from basic algebra and geometry to more advanced calculus and physics. While finding the square root of a perfect square (like 25 or 100) is straightforward, dealing with non-perfect squares like 84 presents a slightly more complex challenge. This article will explore how to find the square root of 84, addressing common hurdles and providing a comprehensive understanding of the process. We'll delve into both approximate and exact methods, ensuring you gain a solid grasp of this crucial mathematical concept.

1. Understanding Square Roots

Before diving into the specifics of finding the square root of 84, let's refresh our understanding of what a square root represents. The square root of a number (x) is a value that, when multiplied by itself, equals x. For example, the square root of 9 is 3, because 3 × 3 = 9. This is denoted as √9 = 3.

However, not all numbers have whole number square roots. Numbers like 84 are non-perfect squares, meaning their square roots are irrational numbers – numbers that cannot be expressed as a simple fraction. This means their decimal representation goes on forever without repeating.

2. Estimating the Square Root of 84

A good starting point when dealing with non-perfect squares is to estimate the value. We can do this by considering the nearest perfect squares. The perfect squares closest to 84 are 81 (9²) and 100 (10²). Since 84 is between 81 and 100, its square root must lie between 9 and 10. This provides a reasonable initial approximation.

To refine this estimate, we can use linear interpolation. The difference between 84 and 81 is 3, and the difference between 100 and 81 is 19. Therefore, the square root of 84 is approximately 9 + (3/19) ≈ 9.16. This is a fairly close estimate.

3. Using a Calculator for a Precise Decimal Approximation

The most practical way to find a precise decimal approximation of the square root of 84 is to use a calculator. Most calculators have a square root function (usually denoted as √ or √x). Simply input 84 and press the square root button. You'll obtain a value close to 9.1651513899... Note that this is still an approximation, as the actual value is an irrational number with an infinite decimal expansion.

4. Finding the Prime Factorization (for exact representation)

While we cannot express the square root of 84 as a simple fraction, we can simplify it using prime factorization. Prime factorization involves breaking down a number into its prime number components.

84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7 = 2² × 3 × 7

Therefore, we can express the square root of 84 as:

√84 = √(2² × 3 × 7) = √2² × √3 × √7 = 2√21

This representation, 2√21, is the simplified exact form of the square root of 84. It avoids decimal approximations and precisely captures the mathematical essence of the root.

5. Applying the Square Root in Practical Contexts

Understanding square roots is crucial in various applications. For example, in geometry, finding the length of the diagonal of a rectangle requires using the Pythagorean theorem, which involves square roots. Similarly, in physics, calculating the velocity or distance often requires working with square roots. The ability to estimate or precisely calculate square roots is therefore vital for effective problem-solving in numerous scientific and engineering disciplines.

Summary

Finding the square root of 84 involves understanding the concept of square roots, estimating the value based on nearby perfect squares, and utilizing a calculator for a precise decimal approximation. However, the most mathematically precise representation is the simplified radical form, 2√21. This form avoids the limitations of decimal approximations, which are inherently imprecise for irrational numbers. Mastering these methods is crucial for tackling diverse mathematical problems across various fields.

FAQs:

1. Is there a method to calculate the square root manually without a calculator? Yes, there are several manual methods like the Babylonian method or the digit-by-digit method, but they are iterative processes requiring multiple steps. For simple approximations, estimation based on nearby perfect squares is often sufficient.

2. Why is the square root of 84 irrational? Because its prime factorization contains prime factors raised to odd powers (3 and 7). A number's square root is rational only if all its prime factors in the factorization have even exponents.

3. Can I use the square root of 84 in further calculations? Absolutely! You can either use the decimal approximation (with its inherent imprecision) or the simplified radical form (2√21) for greater accuracy in subsequent calculations.

4. What is the difference between the approximate and exact value of √84? The approximate value is a decimal representation (e.g., 9.165), while the exact value is the simplified radical form (2√21), which precisely represents the actual mathematical value without any rounding or truncation.

5. How do I know which method to use (estimation, calculator, or simplification)? The choice depends on the context. For quick estimations, use approximation. For precise calculations requiring high accuracy, use a calculator. For mathematical elegance and avoiding decimal approximations in further calculations, use the simplified radical form.

Search Results:

What are vanilla diamonds? - Answers 24 Sep 2023 · What is the square root of 84.5 in inches? Where are the positive rational number placed on a horizontal number line? What are the multiples of 7 up to 500?

How do I graph the equation 16x^2-9y^2+32x+18y-137=0 on a TI … 6 Sep 2014 · First you would have to complete the square on both variables, x and y: 16x^2−9y^2+32x+18y−137=0 Group together the x terms and y terms, and move the constant …

What is the square root of 42? + Example - Socratic 17 Sep 2016 · sqrt(42) ~~ 8479/1350 = 6.48bar(074) ~~ 6.4807407 42=2*3*7 has no square factors, so sqrt(42) cannot be simplified. it is an irrational number between 6 and 7 Note that …

How do you find the square root of 36? - Socratic sqrt(36) = sqrt(6 * 6) = 6. 30451 views around the world You can reuse this answer

Where does the cent sign go before or after the number? 15 Sep 2023 · How do you type a number squared into a square root equation? if you have a TI-84 you can hit math, then go down to the x before the square root sign

What else are suicide sprints called? - Answers 19 Nov 2022 · What expression is equivalent to 84 16? Square root 18 2 square root 75 square root 48? What does the roman numeral S stand for?

Where is the roots button on ti-84 plus? - Answers The square root button is about the "x2" button. Press 2ND and then x2 to use it. If you want a cubed root, press MATH and select 3√(. If you want any other root, press MATH and selcect x√.

How many times does the word king appear in Revelation? 26 Oct 2022 · What is the square root of 4 plus the square root of 9? How many yards of 59 inches material would 84 yards of 45 inches convert to? What controls a slope of a line?

How do you fill out af form 2005? - Answers 8 Apr 2025 · What is the equivalent to square root of fourteen multipled by 10? What is the volume of a cone with a slant length of 130mm and a base diameter of 10cm showing work? What is …

Where is the ans key on a TI-84 plus? - Answers 15 Dec 2022 · What is the square root of 18 plus square root of 32? ... One can purchase a TI 84 Plus Silver Edition from the following shops: Amazon, Calculators Direct, eBay, Walmart, …