Question

If $m - \dfrac{1}{m} = 5$; find the value of ${m^2} - \dfrac{1}{{{m^2}}}$.

Answer 1

Hint: As we have to find the value of ${m^2} - \dfrac{1}{{{m^2}}}$. First, find the value of $m + \dfrac{1}{m}$ by the formula, ${\left( {a + b} \right)^2} = {\left( {a - b} \right)^2} + 4ab$. After that, use the formula, $\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right)$ and substitute the values to get the value of ${m^2} - \dfrac{1}{{{m^2}}}$.

Complete step-by-step solution:
Given: - $m - \dfrac{1}{m} = 5$
As we know,
${\left( {a + b} \right)^2} = {\left( {a - b} \right)^2} + 4ab$
Substitute $a = m$ and $b = \dfrac{1}{m}$. Then,
$ \Rightarrow {\left( {m + \dfrac{1}{m}} \right)^2} = {\left( {m - \dfrac{1}{m}} \right)^2} + 4 \times m \times \dfrac{1}{m}$
Cancel out the common factors,
$ \Rightarrow {\left( {m + \dfrac{1}{m}} \right)^2} = {\left( {m - \dfrac{1}{m}} \right)^2} + 4$
Now substitute the values,
$ \Rightarrow {\left( {m + \dfrac{1}{m}} \right)^2} = {\left( 5 \right)^2} + 4$
Square the term on the right side,
$ \Rightarrow {\left( {m + \dfrac{1}{m}} \right)^2} = 25 + 4$
Now add the terms,
$ \Rightarrow {\left( {m + \dfrac{1}{m}} \right)^2} = 29$
Taking square root on both sides, we get,
$ \Rightarrow m + \dfrac{1}{m} = \sqrt {29} $
We know that,
${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$
Substitute $a = m$ and $b = \dfrac{1}{m}$. Then,
$ \Rightarrow {m^2} - \dfrac{1}{{{m^2}}} = \left( {m - \dfrac{1}{m}} \right)\left( {m + \dfrac{1}{m}} \right)$
Substituting the values, we get,
$ \Rightarrow {m^2} - \dfrac{1}{{{m^2}}} = 5 \times \sqrt {29} $
Multiply the terms on the right side,
$\therefore {m^2} - \dfrac{1}{{{m^2}}} = 5\sqrt {29} $

Hence, the value of ${m^2} - \dfrac{1}{{{m^2}}}$ is $5\sqrt {29} $.

Additional Information: According to the definition of rational number, we can say that the rational numbers are the numbers that can be expressed as a fraction where that fraction can be some positive number, negative number, and zero. These numbers either terminate or repeat after the decimal point when written in decimal form.
According to the definition of irrational numbers, we can say that the irrational numbers are the numbers that are not rational numbers (i.e., those numbers which can never be expressed in the form of $\dfrac{p}{q}$). These numbers have endless non-repeating digits after the decimal point was written in decimal form.

Note: Algebra is a combination of both numerical and letters. Both represent the unknown quantity of the practical situation where a formula is applied to. When the numbers and letters come together with the factorials and matrices, formulas are formed. Algebraic formulas are put to use in particular situations for solving them. One of the most widely used formulas in math is the formula of ${\left( {a + b} \right)^2}$, ${\left( {a - b} \right)^2}$ and $\left( {{a^2} - {b^2}} \right)$.