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Simplify each of the following:
\( 0.76 \times 0.76+2 \times 0.76 \times 0.24+0.24 \times 0.24 \)
Given:
\( 0.76 \times 0.76+2 \times 0.76 \times 0.24+0.24 \times 0.24 \)
To do:
We have to simplify the given expression.
Solution:
We know that,
$(a+b)^2=a^2+b^2+2ab$
$(a-b)^2=a^2+b^2-2ab$
$(a+b)(a-b)=a^2-b^2$
Therefore,
\( 0.76 \times 0.76+2 \times 0.76 \times 0.24+0.24 \times 0.24 \)
The above expression is of the form $a^{2}+2 a b+b^{2}$, where $a=0.76$ and $b=0.24$.
This implies,
$0.76 \times 0.76+2 \times 0.76 \times 0.24+0.24 \times 0.24=(0.76+0.24)^{2}$
$=(1.00)^{2}$
$=1$
Hence, $0.76 \times 0.76+2 \times 0.76 \times 0.24+0.24 \times 0.24=1$.
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