Question

Express the following in exponential form:
\[-\dfrac{5}{7}\times -\dfrac{5}{7}\times -\dfrac{5}{7}\times -\dfrac{5}{7}\]

Answer 1

Hint: We know that an exponential number is a number of the form \[{{a}^{n}}\]. It is also known to us that \[a\times a\times a\ldots ntimes={{a}^{n}}\] which is a fundamental exponential identity, In addition we can write \[-a\] as \[(-1)\times (a)\] which is also an algebraic concept, also \[{{a}^{-n}}=\dfrac{1}{{{a}^{n}}}\]is also applicable in this case. Now all we need to do is to separate the constants and use the above three relations to find exponential representation of the above expression.

Complete step-by-step answer:
We know that \[a\times a\times a\ldots ntimes={{a}^{n}}\]
For example –
  \[3\times 3\times 3\] can be written as \[{{3}^{3}}\]
\[4\times 4\times 4\times 4\times 4\times 4\] can be written as \[{{4}^{6}}\]
This way of representing a number is known as exponential form –
It is basically a notation to represent a number which implies that writing \[3\times 3\times 3\] is equivalent to writing \[{{3}^{3}}\].
Similarly we know that, \[-a=(-1)\times (a)\]
Which is fundamental algebraic identity.
Now we solve using above identities,
\[-\dfrac{5}{7}\times -\dfrac{5}{7}\times -\dfrac{5}{7}\times -\dfrac{5}{7}\]
Now, we separate \[-1\] from \[-\dfrac{5}{7}\],
\[-1\times \dfrac{5}{7}\times -1\times \dfrac{5}{7}\times -1\times \dfrac{5}{7}\times -1\times \dfrac{5}{7}\]
We club the similar terms together so that we can easily count the power to the constant easily and accurately,
\[{{(-1)}^{4}}\times {{\left( \dfrac{5}{7} \right)}^{4}}\]
We can find value of \[{{(-1)}^{4}}\] by using following relation which is –
\[{{(-1)}^{m}}=1\] When m is even number
e.g. \[{{(-1)}^{4}}=-1\times -1\times -1\times -1\] we observe that even number of \[-1\] gets cancelled,
From this fact we can deduce that -1 raised to the power n is 1 if n is even or -1 if n is odd, this is an important relation usually used in algebra.
\[{{\left( \dfrac{5}{7} \right)}^{4}}\]
\[{{5}^{4}}\times {{7}^{-4}}\]

Note: The common mistakes committed by students include leaving answers in fractional form, forgetting about \[-1\] and writing it as it is in the answer. In the example student can use \[{{x}^{n}}\] in place \[{{a}^{n}}\] because change of variable doesn’t affect expression. In addition to it students must remember all fundamental formulas required to solve the above question.