Question

# Square of $- \dfrac{{13}}{{17}}$ is …………………..

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Hint: To find the square of a real number, we need to multiply the number with itself. Since, ${n^2} = n \times n$ , where n is a real number. Also, the whole square of a negative real number is always positive since a negative real number multiplied with a negative real number is a positive, as per the algebraic rules of multiplication.

According to the question we need to find the value of the square of $- \dfrac{{13}}{{17}}$. So, to find the value of square of $- \dfrac{{13}}{{17}}$
Let us assume that $n = - \dfrac{{13}}{{17}}$ ,
Then to find the square of $- \dfrac{{13}}{{17}}$ , we need to find the value of ${n^2}$ with $n = - \dfrac{{13}}{{17}}$.
Then, ${n^2} = n \times n$will become:
${n^2} = n \times n$
Now, putting $n = - \dfrac{{13}}{{17}}$we will get ${n^2}$ as:
$\Rightarrow {\left( { - \dfrac{{13}}{{17}}} \right)^2} = \left( { - \dfrac{{13}}{{17}}} \right) \times \left( { - \dfrac{{13}}{{17}}} \right) \\ \Rightarrow {\left( { - \dfrac{{13}}{{17}}} \right)^2} = \dfrac{{169}}{{289}} \\$
That is, the square of $- \dfrac{{13}}{{17}}$will be $\dfrac{{169}}{{289}}$ a positive real number. Since as per the algebraic laws of multiplication a negative real number when multiplied with another negative real number gives a positive real number since:
$\left( - \right) \times \left( - \right) = \left( + \right)$
Thus, the sign of the product of a minus with a minus will always be a plus sign.
Therefore, the sign of $\dfrac{{169}}{{289}}$will be plus.
Therefore, Square of $- \dfrac{{13}}{{17}}$ is ………$\dfrac{{169}}{{289}}$…………..
Note: The cube of the real number is calculated as the number multiplied with itself thrice. If the number is $n,$ then ${n^3} = n \times n \times n$ . The cube of a negative real number will always be negative because :
${\left( { - n} \right)^3} = \left( { - n} \right) \times \left( { - n} \right) \times \left( { - n} \right) \\ \Rightarrow {\left( { - n} \right)^3} = \left( {{n^2}} \right)\left( { - n} \right) \\ \Rightarrow {\left( { - n} \right)^3} = - {\left( n \right)^3} \\$

Note: A positive number when multiplied with a negative number will always result in a negative number.
Also, for a positive real number we do not need to put the plus sign additionally, because no sign before a real number conventionally means the number is positive.