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How do you write 5 ( to the power of 8 ) as a quotient of 2 exponential terms with the same base in 4 different ways by using only positive non zero exponents ?

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Last updated date: 26th Feb 2024
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IVSAT 2024
Answer
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Hint:The Question says about the Quotient rule of exponents which states that if we divide two exponents with the same base , we just then keep the base and subtract the powers . According to the question base will become our base and the exponent will be 8 and we have to write in the way as the quotient rule says but in 4 different ways by using only positive non zero exponents.

Complete step by step answer:
We will first set the base as 5 and exponent as 8 which must come by applying a quotient exponential rule.First we will find the four combinations of numbers such that the difference of both the exponents can be calculated as 8.By applying Quotient exponential rule – Here $x$ and $y$ are the exponents and their difference must be 8 while applying Quotient exponential rule .
$ \dfrac{{{5^x}}}{{{5^y}}}$
$\Rightarrow x - y = 8$

Now let us make the 4 pairs of combination such that it gives 8 when the powers with same base are subtracted like the following in different ways –
The numbers 9 and 1 make the difference of the number 8 as we wanted.
The numbers 10 and 2 make the difference of the number 8 as we wanted.
The numbers 15 and 7 make the difference of the number 8 as we wanted.
The numbers 1000 and 992 make the difference of the number 8 as we wanted .
We can write these as =>
$\dfrac{{{5^9}}}{{{5^1}}} \\
\Rightarrow\dfrac{{{5^{10}}}}{{{5^2}}} \\
\Rightarrow\dfrac{{{5^{15}}}}{{{5^7}}} \\
\therefore\dfrac{{{5^{1000}}}}{{{5^{992}}}}$

Note: Avoid negative exponents for easier calculations.If there is more than one term in brackets , with an power outside the brackets then the power is distributed to every term in the brackets.The variable in the denominator , numerator and exponent cannot be equal to 0.Through quotient rule we can simplify an algebraic expression with expressions.