Using A Suitable Identity, Find The Product Of The Following (5x-9y) 2 $ $ ​Simplify: (x + 𝟏 𝒙 ) (x - 𝟏 𝒙 ) (π’™πŸ + 𝟏 π’™πŸ ) (π’™πŸ’ + 𝟏 π’™πŸ’)

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Introduction

In algebra, identities are used to simplify complex expressions and make them easier to work with. One of the most common identities used in algebra is the difference of squares identity, which states that (a + b)(a - b) = a^2 - b^2. In this article, we will use this identity to simplify the given expression (5x - 9y)^2 and then use another identity to simplify the expression (x + 1x)(x - 1x)(x^2 + 1x2)(x4 + 1x^4).

Simplifying the First Expression

To simplify the expression (5x - 9y)^2, we will use the difference of squares identity. This identity states that (a + b)(a - b) = a^2 - b^2. In this case, we can rewrite the expression as (5x)^2 - (9y)^2.

(5x - 9y)^2 = (5x)^2 - (9y)^2

Now, we can simplify each term separately. The square of 5x is 25x^2, and the square of 9y is 81y^2.

(5x)^2 = 25x^2
(9y)^2 = 81y^2

Now, we can substitute these values back into the original expression.

(5x - 9y)^2 = 25x^2 - 81y^2

Simplifying the Second Expression

To simplify the expression (x + 1x)(x - 1x)(x^2 + 1x2)(x4 + 1x^4), we will use the distributive property and the identity (a + b)(a - b) = a^2 - b^2.

(x + 1x)(x - 1x) = x^2 - (1x)^2
(x^2 + 1x^2)(x^4 + 1x^4) = x^6 + x^8 - (1x^2)^2 - (1x^4)^2

Now, we can simplify each term separately. The square of x is x^2, and the square of 1x is (1x)^2 = x^2.

(x^2 - (1x)^2) = x^2 - x^2
(x^6 + x^8 - (1x^2)^2 - (1x^4)^2) = x^6 + x^8 - x^4 - x^8

Now, we can substitute these values back into the original expression.

(x + 1x)(x - 1x)(x^2 + 1x^2)(x^4 + 1x^4) = (x^2 - x^2)(x^6 + x^8 - x^4 - x^8)

Simplifying the Final Expression

Now, we can simplify the final expression by combining like terms.

(x^2 - x^2)(x^6 + x^8 - x^4 - x^8) = 0(x^6 - x^4)

The final expression is 0(x^6 - x^4), which is equal to 0.

Conclusion

In this article, we used the difference of squares identity to simplify the expression (5x - 9y)^2 and then used another identity to simplify the expression (x + 1x)(x - 1x)(x^2 + 1x2)(x4 + 1x^4). We found that the final expression is equal to 0.

References

Further Reading

Q: What is the difference of squares identity?

A: The difference of squares identity is a fundamental concept in algebra that states that (a + b)(a - b) = a^2 - b^2. This identity can be used to simplify complex expressions and make them easier to work with.

Q: How do I use the difference of squares identity to simplify an expression?

A: To use the difference of squares identity, you need to identify the two terms in the expression that are being multiplied together. Then, you can rewrite the expression as (a + b)(a - b) and simplify it using the identity.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that states that a(b + c) = ab + ac. This property can be used to simplify complex expressions and make them easier to work with.

Q: How do I use the distributive property to simplify an expression?

A: To use the distributive property, you need to identify the term that is being multiplied by the expression inside the parentheses. Then, you can distribute the term to each term inside the parentheses and simplify the expression.

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change, while a constant is a value that does not change. In algebra, variables are often represented by letters such as x and y, while constants are represented by numbers.

Q: How do I simplify an expression with variables and constants?

A: To simplify an expression with variables and constants, you need to combine like terms and use the rules of algebra to simplify the expression. For example, if you have the expression 2x + 3y, you can combine the like terms 2x and 3y to get 2x + 3y.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when simplifying an expression. The order of operations is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate exponents next.
  3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
  4. Addition and Subtraction: Evaluate addition and subtraction operations from left to right.

Q: How do I use the order of operations to simplify an expression?

A: To use the order of operations, you need to follow the rules of the order of operations. For example, if you have the expression 2x^2 + 3y - 4, you would first evaluate the expression inside the parentheses (if any), then evaluate the exponents, then evaluate the multiplication and division operations from left to right, and finally evaluate the addition and subtraction operations from left to right.

Q: What are some common algebraic identities?

A: Some common algebraic identities include:

  • Difference of squares: (a + b)(a - b) = a^2 - b^2
  • Sum of squares: (a + b)^2 = a^2 + 2ab + b^2
  • Difference of cubes: (a + b)(a^2 - ab + b^2) = a^3 - b^3
  • Sum of cubes: (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Q: How do I use algebraic identities to simplify an expression?

A: To use algebraic identities, you need to identify the identity that applies to the expression you are simplifying. Then, you can use the identity to simplify the expression.

Conclusion

In this article, we have answered some frequently asked questions on simplifying algebraic expressions. We have covered topics such as the difference of squares identity, the distributive property, variables and constants, the order of operations, and common algebraic identities. We hope that this article has been helpful in answering your questions and providing you with a better understanding of simplifying algebraic expressions.