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If ${{5}^{3x}}=125$ and ${{10}^{y}}=0.001$, then find the value of \[x+y\].

Answer
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Hint: Write the given equation in exponent form using the rules of indices which state that when bases are the same the exponents are equal. Then use the value of indices to find the unknown.

Complete step-by-step answer:
In this question we are given that
${{5}^{3x}}=125$ and ${{10}^{y}}=0.001$

If a number is in the form of ${{a}^{b}}$, then a is called base and b is called exponent.
So, to find the value of ‘x’ we will write 125 such as the base will be 5.
We will write $125={{5}^{3}}$.

Substituting the value of 125 in the given equation, we get
${{5}^{3x}}={{5}^{3}}$

Now we will apply rules of indices which state that when bases are the same the exponents are equal.

By comparing ${{5}^{3x}}$ by ${{5}^{3}}$.
We get that,
3x=3
This makes it clear that x=1.

Now we will find the value of ‘y’.
We will write 0.001 such as the base will be 10.
We will write $0.001={{10}^{-3}}$.
Substituting this value in the given equation, we get
${{10}^{y}}={{10}^{-3}}$

Now we will apply rules of indices that are when bases are the same the exponents are equal.
We get that y=-3

In the question we have been given to find,
\[x+y\]
Substituting the obtained values of ‘x’ and ‘y’, we get
$x+y=(1+(-3))=-2$
So, the value of \[x+y\] is ‘-2’.

Note: Students must know the rules of indices by heart to solve such problems with ease.
Another approach to this problem is using the logarithm. In that case also we will get the same result.