What is an Application of Identities?

S3-SA1-0299

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

An application of identities means using special algebraic equations (identities) to solve problems faster and easier. These identities are like shortcuts that are always true for any value of the variables involved. We use them to simplify calculations, especially when dealing with multiplication of large numbers or complex expressions.

Simple Example

Quick Example

Imagine you need to calculate (100 + 2)^2. Instead of doing (102 * 102), which takes time, you can use the identity (a + b)^2 = a^2 + 2ab + b^2. Here, a=100 and b=2. So, it becomes 100^2 + 2*100*2 + 2^2 = 10000 + 400 + 4 = 10404. Much quicker!

Worked Example

Step-by-Step

Let's use an identity to calculate 98^2.

Step 1: Recognize that 98 can be written as (100 - 2).
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Step 2: Choose the appropriate identity. Since we have (a - b)^2, we use the identity (a - b)^2 = a^2 - 2ab + b^2.
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Step 3: Identify 'a' and 'b'. Here, a = 100 and b = 2.
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Step 4: Substitute the values of 'a' and 'b' into the identity: (100 - 2)^2 = 100^2 - 2*(100)*(2) + 2^2.
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Step 5: Perform the calculations: 100^2 = 10000, 2*100*2 = 400, 2^2 = 4.
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Step 6: Combine the results: 10000 - 400 + 4.
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Step 7: Calculate the final answer: 9600 + 4 = 9604.

So, 98^2 = 9604.

Why It Matters

Understanding identities is crucial for solving complex problems in various fields. Engineers use them to design structures, computer scientists apply them in algorithms, and data scientists use them for efficient data processing. It's a foundational skill for careers in technology and research, helping you solve problems faster and more accurately.

Common Mistakes

MISTAKE: Confusing (a + b)^2 with a^2 + b^2 | CORRECTION: Remember that (a + b)^2 expands to a^2 + 2ab + b^2. The '2ab' term is often forgotten.

MISTAKE: Incorrectly applying signs, especially in (a - b)^2. Forgetting the minus sign for the middle term. | CORRECTION: The identity is (a - b)^2 = a^2 - 2ab + b^2. The middle term is negative, but the last term (b^2) is always positive.

MISTAKE: Using identities for expressions that don't fit the pattern, like (a + b + c)^2 for (a + b)^2. | CORRECTION: Only use a specific identity when the expression exactly matches its form. For more terms, there are different identities or you need to expand manually.

Practice Questions

Try It Yourself

QUESTION: Use an identity to calculate 103^2. | ANSWER: 10609

QUESTION: Find the value of 105 * 95 using an identity. (Hint: Think of (a+b)(a-b)) | ANSWER: 9975

QUESTION: If x + y = 7 and xy = 10, find the value of x^2 + y^2. (Hint: Use (x+y)^2) | ANSWER: 29

MCQ

Quick Quiz

Which identity would you use to calculate 52 * 48?

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

(a - b)^2 = a^2 - 2ab + b^2

(a + b)(a - b) = a^2 - b^2

None of the above

The Correct Answer Is:

52 * 48 can be written as (50 + 2)(50 - 2), which perfectly matches the (a + b)(a - b) = a^2 - b^2 identity. This allows for quick calculation as 50^2 - 2^2.

Real World Connection

In the Real World

Identities are used in computer programming to optimize calculations, making apps run faster. For example, when an online shopping app like Flipkart or Myntra needs to process millions of transactions or calculate discounts quickly, efficient mathematical operations using identities can speed up the process and improve user experience.

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all possible values of its variables | ALGEBRAIC EXPRESSION: A combination of variables, numbers, and operation symbols | SIMPLIFY: To make an expression or calculation easier or shorter | VARIABLE: A symbol (like x or y) that represents a quantity that can change | COEFFICIENT: A number multiplied by a variable in an algebraic term

What's Next

What to Learn Next

Next, you should explore factorization of algebraic expressions. Understanding identities makes factorization much easier, as it often involves recognizing patterns that are reverse applications of identities. Keep practicing to become a math pro!