Question

How do you simplify $\left( 4x+9y \right)\left( 4x-9y \right)$ ?

Answer 1

Hint: In this problem, we have to apply the distributive property two times. At first, we apply it to multiply the first bracket to the terms of the second bracket. After that, we again apply it to break down the further brackets. After that, we cancel off the like terms and then get our final expression simplified.

Complete step by step solution:
The given expression that we have at our disposal is,
$\left( 4x+9y \right)\left( 4x-9y \right)$
As the above expression involves the multiplications of two expressions within brackets, we use the distributive property. The distributive property states that an expression of the form $a\left( c+d \right)$ can be written as $ac+ad$ . Here, $a=\left( 4x+9y \right),c=4x,d=-9y$ . Applying this to the above expression, the expression thus becomes,
$\Rightarrow \left( 4x+9y \right)\times 4x+\left( 4x+9y \right)\times \left( -9y \right)$
Now, we again apply the distributive property to the first group of the above expression. Then, $a=4x,c=4x,d=9y$ . Applying this to the above expression, the expression thus becomes,
$\Rightarrow \left( 4x\times 4x+9y\times 4x \right)+\left( 4x+9y \right)\times \left( -9y \right)$
Now, we again apply the distributive property to the second group of the above expression. Then, $a=4x,c=4x,d=9y$ . Applying this to the above expression, the expression thus becomes,
$\Rightarrow \left( 4x\times 4x+9y\times 4x \right)+\left( 4x\times \left( -9y \right)+9y\times \left( -9y \right) \right)$
Multiplying the terms within the brackets, we get,
$\Rightarrow \left( 16{{x}^{2}}+36yx \right)+\left( -36xy-81{{y}^{2}} \right)$
Opening up the brackets in the above expression , the above expression thus becomes,
$\Rightarrow 16{{x}^{2}}+36yx-36xy-81{{y}^{2}}$
Since, in the above expression there are two like terms with two different signs, we can cancel them off. Doing this, the equation thus becomes,
$\Rightarrow 16{{x}^{2}}-81{{y}^{2}}$
Therefore, we can conclude that the given expression can be simplified to $16{{x}^{2}}-81{{y}^{2}}$ .

Note:
As this problem requires application of the distributive property three times, we have to be careful while handling so many terms at the same time. We should take care of the signs and simplify the expression to the maximum possible in each step to avoid errors. Also, there is a predefined formula for these type of expressions, which is $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$ . We can apply this directly.