Applications of Identities: A mathematical identity is a relationship between two mathematical expressions in variables like \(a\) and \(b\). Algebraic identities are algebraic equations that are valid for all values of variables in them. Algebraic identities are used in this way to calculate algebraic expressions and help solve polynomials. 

Algebraic identities play an important role in the study of mathematics in general. The fundamental algebraic identities are primarily useful in solving a variety of mathematics problems. It will also help to improve their proficiency in using these methods in algebraic manipulations and problem-solving.
This article helps a lot in understanding algebraic identities and their applications.

Algebraic Identities

Algebraic identities are a set of formulas used in mathematics. They are the basic working principle of algebra, and they are useful for performing computations in simple steps. To solve some algebraic problems, we will need to work through a series of mathematical steps. We can perform the calculations without doing any extra steps. The binomial expansion of terms has produced several algebraic identities. The polynomial factorisation is also done with them. 

For any values of the variables, an algebra identity means that the left-hand side of the equation is identically equal to the right-hand side. We’ll try to identify ourselves with the algebraic identities, their proofs, and how to use them in our math calculations here.

Types of Algebraic Identities

A list of the algebraic identities is given below.

1. \((a+b)^{2}=a^{2}+2 a b+b^{2}\)
2. \((a-b)^{2}=a^{2}-2 a b+b^{2}\)
3. \(a^{2}-b^{2}=(a+b)(a-b)\)
4. \((x+a)(x+b)=x^{2}+(a+b) x+a b\)
5. \((x+y+z)^{2}=x^{2}+y^{2}+z^{2}+2 x y+2 y z+2 x z\)
6. \((a+b)^{3}=a^{3}+b^{3}+3 a b(a+b)\)
7. \((a-b)^{3}=a^{3}-b^{3}+3 a b(a-b)\)
8. \(a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c)\left(a^{3}+b^{3}+c^{3}-a b-b c-c a\right)\)
   or
   \(a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c)(a+b+c)^{2}-3(a b+b c+c a)\)
   If \((a+b+c)=0\) then \(\left(a^{3}+b^{3}+c^{3}\right)=3 a b c\)
9. \(\left(a^{3}+b^{3}\right)=(a+b)\left(a^{2}-a b+b^{2}\right)\)
10. \(\left(a^{3}-b^{3}\right)=(a+b)\left(a^{2}+a b+b^{2}\right)\)

Use of Identities in Mathematics

An identity is a mathematical equation that holds regardless of the variables’ values. They’re used to rewrite or simplify algebraic expressions. Because the two sides of identity are interchangeable by definition, we can change one for the other at any time.

Some other uses of identities are listed below.

1. The algebraic identities can be used to solve squares and products of algebraic expressions. 
2. The algebraic identities are also used to calculate products of numbers and other things.
3. Algebraic Identities help while doing factorisation
4. Algebraic Identities help in simplification, and it helps to solve faster way.
5. Algebraic Identities play a major role in solving quadratic equations and higher power equations.
6. Algebraic Identities helps in proving trigonometric identities
7. Algebraic Identities while solving integration and differentiation problems.

Examples of Applications of Identities

Let us see how we can apply identities to solve various problems.

Example -1: Evaluate \((103)^{3}\).

We are asked to find the cube of \(103\).

We can write \(103\) as \(100+3\).

So, \((103)^{3}\) can be written as \((103)^{3}=(100+3)^{3}\)

This can be solved by applying the algebraic identity

\((a+b)^{3}=a^{3}+b^{3}+3 a b(a+b)\)

\(\Rightarrow(100+3)^{2}=100^{3}+3^{3}+3(100)(3)(100+3)\)

\(=1000000+27+900(103)=1092727\)

\(\Rightarrow(103)^{3}=1092727\)

Example -2: Evaluate \((97)^{2}\).

We are asked to find the square of \(97\).

We can write \(97\) as \(100-3\).

So, \((97)^{2}\) can be written as \((97)^{2}=(100-3)^{2}\)

The given can be solved by applying the algebraic identity

\((a-b)^{2}=a^{2}-2 a b+b^{2}\)

\(\Rightarrow(100-3)^{2}=100^{2}-2(100)(3)+3^{2}\)

\(=10000-600+9\)

\(\Rightarrow(97)^{2}=9409\)

Example-3: Use the suitable identity to solve \((3 x+4)(3 x-4)\).

Here, a binomial and its conjugate is given to multiply.

The given can be solved by applying the algebraic identity

\((a+b)(a-b)=a^{2}-b^{2}\)

\(\Rightarrow(3 x+4)(3 x-4)=(3 x)^{2}-4^{2}\)

\(=9 x^{2}-16\)

Therefore, \((3 x+4)(3 x-4)=9 x^{2}-16\).

Example-4: Solve \(402 \times 403\)

Here, we are asked to find the product of two numbers.

We can write \(402\) as \(400+2\) and \(403\) as \(400+3\).

So, the given product can be written as \(402 \times 403=(400+2) \times(400+3)\)

Use the Identity \((x+a)(x+b)=x^{2}+(a+b) x+a b\)

Where \(x=400, a=2, b=3\)

\(402 \times 403=(400+2) \times(400+3)\)

\(=400^{2}+(2+3) 400+2 \times 3\)

\(=160000+2000+6\)

\(=162006\)

Therefore, \(402 \times 403=162006\).

Example-5: Evaluate \((107)^{3}\)

Given \((107)^{3}\).

The given can be written as \((107)^{3}=(100+7)^{3}\)

The given can be solved by applying the algebraic identity

\((a+b)^{3}=a^{3}+b^{3}+3 a b(a+b)\)

\(\Rightarrow(100+7)^{3}=100^{3}+7^{3}+3(100)(7)(100+7)\)

\(=1000000+343+(2100)(107)\)

\(=1000343+224700\)

Therefore, \((107)^{3}=1,225,043\)

Solved Examples – Applications of Identities

Q.1. Solve by using the suitable algebraic identity: \(298 \times 302\).
Ans: We are asked to find the product of numbers \(298 \,\&\, 302\).
\(298 \times 302\) can be written as \((300-2) \times(300+2)\)
We can use the algebraic identity \(a^{2}-b^{2}=(a+b)(a-b)\) to solve the given.
Here we have \(a=300\), and \(b=2\)
Substituting the values in the above identity, we get:
\((300-2)(300+2)=300^{2}-2^{2}\)
\(=90000-4\)
\(=89996\)
Therefore, \(298 \times 302=89996\).

Q.2. The area of a square is \(16 x^{2}+16 x+4\). What is the measure of the side of the square?
Ans: Given: \(16 x^{2}+16 x+4\)
By observing the given expression, we can see that it can be simplified using the algebraic identity \((a+b)^{2}=a^{2}+2 a b+b^{2}\)
Comparing this with the given expression \(16 x^{2}+16 x+4\), we have \(a^{2}=16 x^{2} \Rightarrow a=4 x\), \(b^{2}=4 \Rightarrow b=2\)
\(16 x^{2}+16 x+4=4 x^{2}+2(4 x)(2)+2^{2}\)
\(\Rightarrow 16 x^{2}+16 x+4=(4 x+2)^{2}\)
Therefore, the area of the square in terms of the product of its sides is the LHS of the identity \((a+b)^{2}\)
Therefore, the side of the square is \((4 x+2)\).

Q.3. Evaluate \((104)^{3}\).
Ans: Given: \((104)^{3}\)
The given can be written as \((104)^{3}=(100+4)^{3}\)
The given can be solved by applying the algebraic identity
\((a+b)^{3}=a^{3}+b^{3}+3 a b(a+b)\)
\((100+4)^{3}=100^{3}+4^{3}+3(100)(4)(100+4)\)
\(\Rightarrow 1000000+64+1200(104)=1124864\)
\(\Rightarrow(104)^{3}=1124864\)

Q.4. Find the product of \((x+2)(x+2)\) using standard algebraic identities.
Ans: Given: \((x+2)(x+2)\)
\((x+2)(x+2)\) can be written as \((x+2)^{2}\)
Thus, it is of the form of identity \((a+b)^{2}=a^{2}+2 a b+b^{2}\)
where \(a=x\) and \(b=2\)
So we have,
\((x+2)^{2}=x^{2}+2(x) 2+2^{2}\)
\(\Rightarrow x^{2}+4 x+4\)
Therefore, \((x+2)(x+2)=x^{2}+4 x+4\)

Q.5. Use the Identity \((x+a)(x+b)=x^{2}+(a+b) x+a b\) to find the following: \(501 \times 502\)
Ans: Given: \(501 \times 502\)
The given can be written as \((500+1) \times(500+2)\)
Use the Identity \((x+a)(x+b)=x^{2}+(a+b) x+a b\)
Where \(x=500, a=1, b=2\)
\(501 \times 502=500^{2}+(1+2) 500+1 \times 2\)
\(=250000+1500+2\)
\(=251502\)
Therefore, \(501 \times 502=251502\).

Q.6. Evaluate \((305)^{3}\) by using the suitable algebraic identity.
Ans: Given \((305)^{3}\).
The given can be written as \((305)^{3}=(300+5)^{3}\)
The given can be solved by applying the algebraic identity
\((a+b)^{3}=a^{3}+b^{3}+3 a b(a+b)\)
\((300+5)^{3}=300^{3}+5^{3}+3(300)(5)(300+5)\)
\(\Rightarrow 27000000+125+4500(305)\)
\(\Rightarrow 27000000+125+1372500\)
\(=2,83,72,625\)
\(\Rightarrow(305)^{3}=2,83,72,625\)

Q.7. Solve by using the suitable algebraic identity: \(198 \times 202\).
Ans: \(298 \times 302\) can be written as \((200-2) \times(200+2)\)
Use the algebraic identity \(a^{2}-b^{2}=(a+b)(a-b)\) to solve the given.
Here we have \(a=200\), and \(b=2\)
Substituting the values in the above identity, we get:
\((200-2)(200+2)=200^{2}-2^{2}\)
\(=40000-4\)
\(=39996\)
Therefore, \(198 \times 202=39996\)

Summary

Identities, which are equations that explain to us how to solve a problem in mathematics, tell us exactly what we need to do to solve it. If we didn’t use identities, we would have to do all of the work that led up to the formula to get our answer. Algebraic identities are very important in the mathematics curriculum and mathematics in general. Students can learn mathematical techniques by knowing and understanding these identities. It will also help them develop fluency using these methods in algebraic manipulations and problem-solving. This article includes the different algebraic identities and their applications.

Frequently Asked Questions (FAQ) – Applications of Identities

Q.1. Why do we use identities?
Ans: They are used to rewrite or simplify algebraic expressions. Because the two sides of identity are interchangeable by definition, we can change one for the other at any time. 

Q.2. What is the use of algebraic identities?
Ans: Algebraic identities are algebraic equations that are valid for all values of variables in them. The polynomial factorisation is also done with them. Algebraic identities are used in this way to calculate algebraic expressions and help solve polynomials.

Q.3. What is standard identity?
Ans: A mathematical identity is a relationship between two mathematical expressions in variables like \(a\) and \(b\). Algebraic identities are algebraic equations that are valid for all values of variables in them.

Q.4. What are the four identities?
Ans: Basic four algebraic identities are,
1. \((a+b)^{2}=a^{2}+2 a b+b^{2}\)
2. \((a-b)^{2}=a^{2}-2 a b+b^{2}\)
3. \(a^{2}-b^{2}=(a+b)(a-b)\)
4. \((x+a)(x+b)=x^{2}+(a+b) x+a b\)

Q.5. Why do we study identities in mathematics?
Ans: Identities in mathematics, which are equations that show us how to solve a problem, tell us exactly what we need to do to solve it. We would have to do all of the work that leads up to the formula to find our answer if we didn’t use formulas.

Q.6. How to verify algebraic identity?
Ans: The substitution method is used to verify the algebraic identities. Substitute the values for the variables and perform the arithmetic operation with this method.

Q.6. List the uses of algebraic identities.
Ans: Some other uses of Identities are listed below.
1. The algebraic identities can be used to solve squares and products of algebraic expressions.
2. The algebraic identities are also used to calculate products of numbers and other things.
3. Algebraic Identities help while doing factorisation
4. Algebraic Identities help in simplification, and it helps to solve faster way.
5. Algebraic Identities play a major role in solving quadratic equations and higher power equations.
6. Algebraic Identities helps in proving trigonometric identities
7. Algebraic Identities while solving integration and differentiation problems.

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