Question

Simplify: \[{(6{x^2} - 5y)^2}\]

Answer 1

Hint:
We will simplify the given polynomial by using an appropriate algebraic identity. We will first apply the identity to the polynomial and find the value of each term individually. We will then use laws of exponents to simplify each term. Finally we will substitute the values of each term in the identity to find the required answer.

Formula used:
We will use the following formulas:
1) \[{(a - b)^2} = {a^2} - 2ab + {b^2}\]
2) \[{\left( {{x^m}} \right)^n} = {x^{mn}}\]
3) \[{\left( {xy} \right)^m} = {x^m}{y^m}\]

Complete step by step solution:
The given polynomial is \[{(6{x^2} - 5y)^2}\]. We have to simplify this polynomial by applying an appropriate identity. We observe that the suitable identity for this case would be \[{(a - b)^2} = {a^2} - 2ab + {b^2}\].
Here, \[a = 6{x^2}\] and \[b = 5y\].
Now, \[{a^2} = {\left( {6{x^2}} \right)^2}\].
Let us apply the law of exponents \[{\left( {xy} \right)^m} = {x^m}{y^m}\] to \[{\left( {6{x^2}} \right)^2}\]. This gives us
\[ \Rightarrow {a^2} = {6^2} \times {\left( {{x^2}} \right)^2} = 36{\left( {{x^2}} \right)^2}\] ……….\[(1)\]
We will now apply the law of exponent \[{\left( {{x^m}} \right)^n} = {x^{mn}}\] to \[{\left( {{x^2}} \right)^2}\]. We get
\[{\left( {{x^2}} \right)^2} = {x^{2 \times 2}} = {x^4}\] ……….\[(2)\]
From equations \[(1)\] and \[(2)\] we get
\[ \Rightarrow {a^2} = 36{x^4}\] ……….\[(3)\]
Now we have to find the value of \[2ab\]. We know that \[a = 6{x^2}\] and \[b = 5y\]. Thus, we get
\[2ab = 2 \times \left( {6{x^2}} \right) \times 5y\]
In the above equation, we should multiply the coefficients, which are 2, 6 and 5. The product of these coefficients is 60. Hence, we have
\[ \Rightarrow 2ab = (2 \times 6 \times 5){x^2}y = 60{x^2}y\] ……….\[(4)\]
Finally, we have to find the value of \[{b^2}\]. We know that \[b = 5y\]. So, we get
\[{b^2} = {\left( {5y} \right)^2}\]
Applying the law of exponent \[{\left( {xy} \right)^m} = {x^m}{y^m}\] to \[{\left( {5y} \right)^2}\], we get
\[ \Rightarrow {b^2} = {5^2} \times {y^2}\]
Thus, we get the value of \[{b^2}\] as
\[ \Rightarrow {b^2} = 25{y^2}\] ……….\[(5)\]
We will use equations \[(3)\], \[(4)\], and \[(5)\] in the identity \[{(a - b)^2} = {a^2} - 2ab + {b^2}\]. Therefore,

\[{(6{x^2} - 5y)^2} = 36{x^4} - 60{x^2}y + 25{y^2}\]

Note:
We can also solve this question as an alternate way.
We will break the expression and rewrite it as:
 \[{\left( {6{x^2} - 5y} \right)^2} = \left( {6{x^2} - 5y} \right)\left( {6{x^2} - 5y} \right)\]
Rewriting the equation, we get
\[ \Rightarrow {\left( {6{x^2} - 5y} \right)^2} = 6{x^2}\left( {6{x^2} - 5y} \right) - 5y\left( {6{x^2} - 5y} \right)\]
Multiplying the terms using the distributive property , we get
\[ \Rightarrow {\left( {6{x^2} - 5y} \right)^2} = 36{x^4} - 30{x^2}y - 30{x^2}y + 25{y^2}\]
Adding the like terms, we get
\[ \Rightarrow {\left( {6{x^2} - 5y} \right)^2} = 36{x^4} - 60{x^2}y + 25{y^2}\]