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Find the value of ${\left( { - 5} \right)^3} \times {\left( { - 5} \right)^6}$=?

Answer
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Hint – In this problem we have to find the value of the given expression. Use the basic exponential properties like ${a^p}{a^q} = {a^{p + q}}$ and the basics of ${\left( { - 1} \right)^n} = 1$ if n is even else if n is odd then it is equal to -1, to reach the solution.

“Complete step-by-step answer:”
Given equation is
${\left( { - 5} \right)^3} \times {\left( { - 5} \right)^6}$
So, we have to calculate the value of this
So, the above equation is also written as,
$ \Rightarrow {\left( { - 1 \times 5} \right)^3} \times {\left( { - 1 \times 5} \right)^6} = {\left( { - 1} \right)^3} \times {\left( 5 \right)^3} \times {\left( { - 1} \right)^6} \times {\left( 5 \right)^6}$
Now as we know ${\left( { - 1} \right)^n} = 1$ if n is even and ${\left( { - 1} \right)^n} = - 1$ if n is odd, so use this property in above equation we have,
So, in the above equation ${\left( { - 1} \right)^3} = - 1$ as 3 is an odd number.
And ${\left( { - 1} \right)^6} = 1$ as 6 is an even number.
Therefore above equation becomes,
$ \Rightarrow {\left( { - 1} \right)^3} \times {\left( 5 \right)^3} \times {\left( { - 1} \right)^6} \times {\left( 5 \right)^6} = \left( { - 1} \right){\left( 5 \right)^3}\left( 1 \right){\left( 5 \right)^6}$
Now as we know that if base is same and the number is in multiplication than power got added for example ${a^p}{a^q} = {a^{p + q}}$, so use this property in above equation we have,
$ \Rightarrow \left( { - 1} \right){\left( 5 \right)^3}\left( 1 \right){\left( 5 \right)^6} = - {\left( 5 \right)^{3 + 6}} = - {\left( 5 \right)^9}$
$ \Rightarrow {\left( { - 5} \right)^3} \times {\left( { - 5} \right)^6} = - {\left( 5 \right)^9}$
So, this is the required answer.

Note – Whenever we face such a type of problem the key point is simply about the understanding of some basic identities. Good gist of exponential properties can surely help in getting on the right track to reach the solution.