Question & Answer

Simplify the given equation : $\dfrac{25\times {{t}^{-4}}}{{{5}^{-3}}\times 10\times {{t}^{-8}}}$.

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Hint: We will understand simplification of powers related to various numbers involved in the problem and then present the solution with least terms possible.

Complete step-by-step answer:
There are several types of operators which are available in mathematics. Most basic types of operators are addition, subtraction, multiplication and division. By performing certain operations using these operators we can simplify an expression to least possible degree associated with respective numbers.
In fractional form simplification simply means that the expression should be presented with least terms free from radicals (powers) if possible.
According to our question we are given the expression: $\dfrac{25\times {{t}^{-4}}}{{{5}^{-3}}\times 10\times {{t}^{-8}}}$
First, rewriting all the numbers in the form of factors:
$\dfrac{5\times 5\times {{t}^{-4}}}{{{5}^{-3}}\times 2\times 5\times {{t}^{-8}}}$
Now, cancelling 5 from numerator and denominator we get,
$\dfrac{5\times {{t}^{-4}}}{{{5}^{-3}}\times 2\times {{t}^{-8}}}$
Now, we try to reduce the powers of each term by using the formula for division of power $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$
So, rearranging the suitable exponent we get,
  & \dfrac{{{5}^{1-(-3)}}\times {{t}^{-4-(-8)}}}{2} \\
 & \Rightarrow \dfrac{{{5}^{1+3}}\times {{t}^{-4+8}}}{2} \\
 & \Rightarrow \dfrac{{{5}^{4}}\times {{t}^{4}}}{2} \\
 & \Rightarrow \dfrac{625\times {{t}^{4}}}{2} \\
So, the simplified expression is: $\dfrac{625\times {{t}^{4}}}{2}$.

Note: Students must be careful while operating the exponent of a number. The sign of the exponent with respect to the number must be correctly noted. By using this methodology, we obtained the simplified form without error.