Question & Answer

Evaluate ${{(49)}^{2}}$ using the identity ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$

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Hint: To evaluate ${{(49)}^{2}}$ using the ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ identity write 49 as 50 - 1 , now compare and substitute the values in the given identity.

Complete step-by-step answer:
We can write ${{(49)}^{2}}$ as ${{\left( 50-1 \right)}^{2}}$
Take a = 50 and b = 1, and replace them in the expansion of the formula for calculating the value in mathematical form.
${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$
\[{{\left( 50-1 \right)}^{2}}={{50}^{2}}+{{1}^{2}}-2(50)(1)\]
The value of ${{\left( 50 \right)}^{2}}$is 2500.
\[{{\left( 50-1 \right)}^{2}}=2500+1-100\]
\[{{\left( 50-1 \right)}^{2}}=2501-100\]
\[{{\left( 50-1 \right)}^{2}}=2401\]
Hence the value of ${{(49)}^{2}}$ using the identity ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ is 2401.

Note: You might get confused the algebraic expansion of ${{(a-b)}^{2}}$ and${{a}^{2}}-{{b}^{2}}$ . The algebraic expansion of ${{(a-b)}^{2}}$ is ${{a}^{2}}-2ab+{{b}^{2}}$ and algebraic expansion of ${{a}^{2}}-{{b}^{2}}$ is$(a+b)(a-b)$. Both the algebraic expansions are not equal.