If a^2-1 a^2 =102, find the value of a-1 a . - Vidyadip

Question

If $\text{a}^2-\frac{1}{\text{a}^2}=102,$ find the value of $\text{a}-\frac{1}{\text{a}}.$

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Answer

We have to find the value of $\text{a}-\frac{1}{\text{a}}$
Given $\text{a}^2-\frac{1}{\text{a}^2}=102$
Using identity $(x - y)^2 = x^2 + y^2 - 2xy$
Here $\text{x}=\text{a},\ \text{y}=\frac{1}{\text{a}}$
$\Big(\text{a}-\frac{1}{\text{a}}\Big)^2=\text{a}^2+\Big(\frac{1}{\text{a}}\Big)^2-2\times\text{a}\times\frac{1}{\text{a}}$
$\Big(\text{a}-\frac{1}{\text{a}}\Big)^2=\text{a}^2+\frac{1}{\text{a}^2}-2\times\not\text{a}\times\frac{1}{\not\text{a}}$
By substituting $\text{a}^2-\frac{1}{\text{a}^2}=102$ we get
$\Big(\text{a}-\frac{1}{\text{a}}\Big)^2=102-2$
$\Big(\text{a}-\frac{1}{\text{a}}\Big)^2=100$
$\Big(\text{a}-\frac{1}{\text{a}}\Big)\Big(\text{a}-\frac{1}{\text{a}}\Big)=10\times10$
$\Big(\text{a}-\frac{1}{\text{a}}\Big)=10$
Hence the value of $\Big(\text{a}-\frac{1}{\text{a}}\Big)$ is $10$.

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