Question

By what number should ${{\left( \dfrac{-2}{3} \right)}^{3}}$ be divided so that the quotient maybe equal to ${{\left( \dfrac{9}{4} \right)}^{-2}}$.

Answer 1

Hint: Assume the number that will work as the divisor be ‘$x$’. Use the properties of exponents and powers and Euclid division algorithm to find the value of $x$. To find the value of $x$, divide ${{\left( \dfrac{-2}{3} \right)}^{3}}$ by the given quotient.

Complete step-by-step answer:
Exponentiation is a mathematical operation, written as ${{a}^{n}}$, involving two numbers, the base $a$ and the exponent or power $n$. When $n$is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, ${{a}^{n}}$ is the product of multiplying $n$ bases. Mathematically,

${{a}^{n}}=a\times a\times a\times a.......n\text{ times}$. Some important properties of exponents are:

(i) ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$

(ii) ${{a}^{m}}\div {{a}^{n}}={{a}^{m-n}}$

(iii) ${{a}^{m}}\times {{b}^{m}}={{\left( a\times b \right)}^{m}}$

(iv) ${{a}^{-n}}=\dfrac{1}{{{a}^{n}}}$

Let us know about the Euclid division algorithm. It states that: $dividend=divisor\times quotient+remainder$

So, if remainder is zero, then, $dividend=divisor\times quotient$. Hence, divisors can be found by dividing the dividend by the given quotient. Generally, this algorithm is used to determine the highest common factor of two numbers.

Now, let us come to the question. We have been given, dividend $={{\left( \dfrac{-2}{3} \right)}^{3}}$, quotient =${{\left( \dfrac{9}{4} \right)}^{-2}}$. We have assumed the divisor =$x$. Therefore using Euclid division algorithm:

$\begin{align}
  & {{\left( \dfrac{-2}{3} \right)}^{3}}=x\times {{\left( \dfrac{9}{4} \right)}^{-2}}+0 \\
 & \dfrac{-8}{27}=x\times {{\left( \dfrac{4}{9} \right)}^{2}} \\
 & \dfrac{-8}{27}=x\times \dfrac{16}{81} \\
 & x=\dfrac{-8\times 81}{27\times 16} \\
 & x=\dfrac{-3}{2} \\
\end{align}$

Hence, ${{\left( \dfrac{-2}{3} \right)}^{3}}$ should be divided with $\left( \dfrac{-3}{2} \right)$ to get the quotient equal to ${{\left( \dfrac{9}{4} \right)}^{-2}}$.

Note: It is important to note that, we have put the value of remainder equal to zero. Here, we have been informed nothing about the remainder, so, we have to assume that the divisor completely divides the dividend or in other words, divisor is a factor of the dividend.