Question

How do you simplify \[{\left( { - \dfrac{5}{3}} \right)^2}\]?

Answer 1

Hint: In the given question, we have been given a fraction. The fraction is raised to some integral power. We have to calculate the value of the fraction raised to that power. We can easily do that by just simply multiplying the fraction by itself as many times as is the value of the power.

Complete step by step answer:
In this question, we are going to use the formula of squaring, which is:
\[{a^2} = a \times a\]
The given fraction in the question is \[{\left( { - \dfrac{5}{3}} \right)^2}\].
So, this fraction is going to be multiplied with itself twice, hence,
\[{\left( { - \dfrac{5}{3}} \right)^2} = - \dfrac{5}{3} \times - \dfrac{5}{3} = \dfrac{{25}}{9} = 2\dfrac{7}{9}\]

Note: In this question, the power is positive. But, if it was negative, then we would have first converted it to a positive power by taking the reciprocal of the fraction and then approaching as normal, or,
\[{\left( {\dfrac{a}{b}} \right)^{ - n}} = {\left( {\dfrac{b}{a}} \right)^n}\]
In the question, there was a proper fraction which was squared. We multiplied it twice because the power on its head (two) means that the number is supposed to be multiplied with itself twice. If there was any other power, then we would have multiplied the fraction with itself that many times. The point where we could make a mistake is not taking care of the sign – if the power is even, then the sign of the result is positive, and vice-versa.