Answer

Verified

400.2k+ views

**Hint:**To simplify the given expression we will multiply the term \[(x - y)\] three times in an algebraic mathematical process. On doing calculations we get the required answer.

**Formula used:**Here we will use the following indices formula:

\[{a^{m + n}} = {a^m}{a^n}\]

**Complete Step by Step Solution:**

We have to simplify \[{(x - y)^3}\] in a formation of addition or subtraction with variables.

Now, the term \[(x - y)\] is up to the power of \[3\].

So, we can rewrite the expression as: \[{(x - y)^{1 + 1 + 1}}\].

Now, we can use the above indices formula.

As the expression given is \[{(x - y)^3}\], it is clearly visible that we can multiply \[(x - y)\] three times to achieve the final expression.

So, we can rewrite the expression as following:

\[ \Rightarrow {(x - y)^3} = (x - y) \times (x - y) \times (x - y).\]

So, we will multiply first two \[(x - y)\]’s then the result of it will multiply by another \[(x - y)\] to get the final answer.

Therefore, the multiplication of \[(x - y) \times (x - y)\] is as following:

\[ \Rightarrow (x - y) \times (x - y) = (x - y) \times x - (x - y) \times y\].

So, after the multiplication of these terms, we get the following expression:

\[ \Rightarrow ({x^2} - xy) - (xy - {y^2})\].

Now, if we take the brackets off from the above expression, we get the following expression:

\[ \Rightarrow {x^2} - xy - xy + {y^2}\].

Now, if we arrange the middle terms and other terms accordingly, we get the following expression:

\[ \Rightarrow {x^2} - 2xy + {y^2}\].

So, we can state that :

\[ \Rightarrow {(x - y)^2} = (x - y) \times (x - y)\]

Therefore, \[{(x - y)^2} = {x^2} - 2xy + {y^2}\].

Now, to get the final value of the given expression\[{(x - y)^3}\], we have to multiply \[{(x - y)^2}\] by \[(x - y)\]

Therefore, we can write the following expression:

\[ \Rightarrow {(x - y)^3} = {(x - y)^2} \times (x - y)\].

So, we can rewrite it as following:

\[ \Rightarrow {(x - y)^3} = (x - y) \times ({x^2} - 2xy + {y^2})\].

So, if we multiply the above terms and split up, we get:

\[ \Rightarrow ({x^2} - 2xy + {y^2}) \times x - ({x^2} - 2xy + {y^2}) \times y\].

Now, if we multiply it further, we get:

\[ \Rightarrow ({x^2} \times x - 2xy \times x + {y^2} \times x) - ({x^2} \times y - 2xy \times y + {y^2} \times y)\].

Now, by doing the further simplification, we get:

\[ \Rightarrow ({x^3} - 2{x^2}y + x{y^2}) - ({x^2}y - 2x{y^2} + {y^3})\].

Now, if we take the brackets off, we get:

\[ \Rightarrow {x^3} - 2{x^2}y + x{y^2} - {x^2}y + 2x{y^2} - {y^3}\].

Now, add or subtract the terms having same degrees, we get:

\[ \Rightarrow {x^3} - 3{x^2}y + 3x{y^2} - {y^3}\].

We can rearrange the above statement in following way:

\[ \Rightarrow {x^3} - {y^3} - 3xy(x - y)\].

**\[\therefore \]The answer of the question is \[{x^3} - {y^3} - 3xy(x - y)\].**

**Note:**Points to remember:

If the power of any expression or term is \[n\], where \[n\] is any natural number, then the expression shall multiply \[n\] numbers of times.

Always expand each term in the bracket by all the other terms in the other brackets, but never multiply two or more terms in the same bracket.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Which are the Top 10 Largest Countries of the World?

Write a letter to the principal requesting him to grant class 10 english CBSE

10 examples of evaporation in daily life with explanations

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE