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:

How do you find the square root of 51.84? - Socratic 16 Sep 2015 · You can write it as:: #sqrt(51.84)=sqrt(5184/100)=sqrt(5184)/sqrt(100)=72/10=7.2#

How do you simplify #sqrt(288)#? - Socratic 21 Jul 2015 · How do you simplify #sqrt(288)#? Algebra Radicals and Geometry Connections Simplification of Radical Expressions. 1 Answer

Calculate the mean-square speed, of krypton atoms in a 21 Dec 2017 · Those expressions must be numerically equal, so it must be that 1/2m< v^2 > =3/2kT We can solve for the mean-square speed: < v^2 > =(3kT)/m where: k is Boltzmann's constant (k_b=1.381xx10^(-23)m^2kg//s^2K) T is the temperature of the gas (in Kelvin) m is the mass of one molecule of the gas (in kg) Then T=22^oC=(273+22)K=295K Given that one mole …

How do you simplify #sqrt(84)#? - Socratic 15 Jan 2016 · 2√21 √84=√4*21 = 2√21 Always check if the number inside the root can be broken down into a perfect square which is also must be a factor. then write as √4*21 then take out the 4. when it is taken out '4', factors out and becomes 2 as in 2√21. this can be done until no perfect squares are left inside the root.

What is the square root of 84? - Socratic 10 Mar 2018 · +-2sqrt21 We can break down sqrt84 into the following: sqrt4*sqrt21 We are able to do this because of the property sqrt(ab)=sqrta*sqrtb Where we can separate the radical into the product of the square root of its factors. 21 and 4 are factors of 84. In sqrt4*sqrt21, we can simplify to get: +-2sqrt21 *NOTE: The reason we have a+-sign is because the square root of 4 can be …

How do you simplify Square root of 24 + Square root of 54? 17 Jun 2018 · As a first step you are looking for squared values that you can 'take outside' the roots. Then seeing what you can do with the remaining roots.

What is the square root of 84? - Socratic 25 Apr 2018 · The square root of #84# is an irrational number a little larger than #9#. A square root of a number #n#, is a number #a# such that #a^2 = n#. Actually every positive number has two square roots, but "the" square root is usually taken to mean the positive one. The two square roots of #84# are denoted #sqrt(84)# and #-sqrt(84)#. Note that:

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 would I simplify square root 48+square root 27? - Socratic 19 Jul 2015 · First try and get some squares out of the root 48=4^2*3->sqrt48=sqrt(4^2)*sqrt3=4sqrt3 27=3^2*3->sqrt27 ...

Is there a calculator that will simplify square roots from a 20 Apr 2015 · I believe any scientific calculator can do this. Just press the sqrt key and enter the decimal number, then press = and you should get the square root as a decimal number. For example, if you want to find the square root of 15.087, you would press the sqrt key and enter 15.087, press = and you get 3.884198759075. You would probably round to 3.8842 due to …