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Sanko - Algebra is Useful: 5000 Practice Problems to Show how Algebraic Identities can be used to Simplify Arithmetic Calculations

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Algebra is Useful: 5000 Practice Problems to Show how Algebraic Identities can be used to Simplify Arithmetic Calculations
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Sanko / Norman
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2021
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Algebra is Useful: 5000 Practice Problems to Show how Algebraic Identities can be used to Simplify Arithmetic Calculations

Norman Sanko

What This Book Is About

Many students who encounter algebra for the first time when they enter middle school think that it is only about letters and operations and identities and factors and equations.

They are partly right. Elementary and intermediate algebra are definitely about all those things, but their applications in other fields - physics, economics, biology, business modeling, and so on - are vast, and growing rapidly.

This book focuses on the application of elementary algebraic identities to simplify arithmetic and numerical calculations.

In the age of the calculator and of calculator apps, tedious computations done by hand are almost a thing of the past - but not entirely. As part of the steps required to solve a long problem in mathematics or in allied fields, long numerical calculations are often encountered. It is tedious to pull out the calculator every single time; indeed, some tests do not allow the usage of calculators, even in this day and age.

In such situations, it is very useful to be able to use algebraic identities to simplify complex computations. And who can forget the fact that you look like a genius when you reduce what looks like a difficult task even for a calculator, to something you can do in your head?

This book focuses on the three basic square identities that students spend the first few months of an introductory algebra course mastering. As you go through the theory, through the solved examples, through the Challenge Section, and through the detailed drills, you will find that it is flexible, and that the difficulty level rises slowly. You will be able to level up till your expertise in the subject matches that which you require in your chosen field of study; you can spend ten hours or a thousand hours on the exercises in this book, according to your requirement.

Future books in this series will explore the applications of other identities from intermediate algebra, including the cubic identities.

I am confident that the tools you learn in this book will stand you in good stead throughout your education, and throughout your career.

Best of luck!

Table of Contents

The Identity (a + b) = (a + b + 2ab): Basic Theory

The algebraic identity:

(a + b) = (a + b + 2ab),

Where a and b can take on any numerical values, is the first identity that is taught in most middle school algebra classrooms.

It is also one of the most useful.

This identity is usually used in simplifying long algebraic expressions, but it can also be very useful in making arithmetic calculations more simple. Lets find out how!

Using this algebraic identity to help with arithmetic calculations

For small values of a and b, the identity does not give us any useful shortcuts in calculation.

Assume a = 5, and b = 2, in the identity above.

The Left Hand Side of the identity = (a + b) = (5 + 2) = 7 = 49.

The Right Hand Side = (a + b + 2ab) = (5 + 2 + 2 x 5 x 2) = 25 + 4 + 20 = 49.

Both calculations are relatively equal in complexity - and they, of course, give us the same result.

However, for certain values of a and b, this identity can drastically reduce the amount of computational work we have to do.

Here is an example:

Let us imagine that we were asked to find the value of 1001 (without using a calculator app).

This can be done by multiplying 1001 by 1001, by hand - but this is quite cumbersome, and errors might happen.

Here is another way:

In the identity (a + b) = (a + b + 2ab), we put a = 1000, and b = 1.

The Left Hand Side of this equation becomes (1000 + 1) = (1001), which is exactly what we want to find.

The Right Hand Side of this equation is (a + b + 2ab) = (1000 + 1 + 2 x 1000 x 1) = (1000000 + 1 + 2000) = 1002001.

You will have noticed that this method was very simple, because we were able to avoid difficult calculations. It is also more likely to be error-free, because all we have to worry about is addition, which is much easier to do by hand than a long multiplication.

The same identity can help us simplify many other cumbersome arithmetic calculations.

In the next chapter, you will see a few different calculations for which you can take the help of our trusty identity. Detailed solutions are also provided, so that you can arm yourself with all the examples you need, before you move to the Challenge Section!

The Identity (a + b) = (a + b + 2ab): Practice Problems

Detailed solutions can be found in the next chapter.

1. Please find the value of 1002, by using algebraic identities to simplify computation.

2. What is the value of 557 + 443 + (557 x 886)? Please use algebraic identities to make the computation simple.

3. What is the value of [4408 - 2244 - 2164] / [4488]?

4. What is the square root of [983 + 182 + 357812]?

Solutions: The Identity (a + b) = (a + b + 2ab): Practice Problems

1. Please find the value of 1002, by using algebraic identities to simplify computation.

Solution: Since 1002 is only a little higher than 1000, we can deduce that the identity (a + b) = (a + b + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 1000 and b = 2. After the substitution, we get 1002 = (1000 + 2) = 1000 + 2 + (2 x 1000 x 2) = 1000000 + 4 + 4000 = 1004004.

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2. What is the value of 557 + 443 + (557 x 886)? Please use algebraic identities to make the computation simple.

Solution: First, we note that 886 = 2 x 443. Using this fact, we can rewrite the given expression as 557 + 443 + (2 x 557 x 443). Comparing this with the Right Hand Side of the identity (a + b) = (a + b + 2ab), we get 557 + 443 + (557 x 886) = 557 + 443 + (2 x 557 x 443) = (557 + 443) = 1000 = 1000000.

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3. What is the value of [4408 - 2244 - 2164] / [4488]?

Solution: First, we look at the three numbers under the square signs in the numerator: 4408, 2244, and 2164. We note that 4408 = 2244 + 2164. We also note, in the denominator, that 4488 = 2 x 2244.

Hence, we can guess that the algebraic identity (a + b) = (a + b + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression.

After some trial and error, we note that [(a + b) - a - b] / 2a = b.

In this rearranged equation, we put a = 2244, and b = 2164, to get [(2244 + 2164) - 2244 - 2164] / 4488 = [4408 - 2244 - 2164] / 4488 = 2164. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 2164.

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4. What is the square root of [983 + 182 + 357812]?

Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b) = (a + b + 2ab). Indeed, if we set a = 983, and b = 182, we see that 2ab = 357812, which is exactly equal to the absolute value of the third term in the expression.

Hence, we can say that [983 + 182 + 357812] = [983 + 182].

Therefore, the square root of the expression given is equal to [983 + 182] = 1165.

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The Identity (a - b) = (a + b - 2ab): Basic Theory

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