Question

What is the value of the expression \[{\left( { - 3} \right)^4}\] ?

Answer 1

Hint:We can see this problem is from indices and powers. This number given is having $ - 3$ as base and $4$ as power. But since we have to simplify the expression, we will first write $ - 3$ as $ - 1 \times 3$ and then distribute the power accordingly. Then, we will find the values separately and multiply the entities to find the result. The result thus obtained will be the answer to the given question.

Complete step by step answer:
So, the given question requires us to simplify $ - 3$ to the power \[4\]. $ - 3$ to the power \[4\] can be written as \[{\left( { - 3} \right)^4}\]. We will first express $ - 3$ as $ - 1 \times 3$ in the expression. So, we get,
\[{\left( { - 3} \right)^4} = {\left( { - 1 \times 3} \right)^4}\]

Now, this is of the form ${\left( {a \times b} \right)^x}$. But we can rewrite the expression using the laws of indices and powers as ${\left( {a \times b} \right)^x} = {a^x} \times {b^x}$.
Thus we will apply same on the question above, we get,
\[ \Rightarrow {\left( { - 3} \right)^4} = {\left( { - 1} \right)^4} \times {3^4}\]

Now, we will solve both the entities separately and then multiply both of them to get the final result. We know that \[\left( { - 1} \right)\] raised to an even power is equal to one, So, we get,
\[ \Rightarrow {\left( { - 3} \right)^4} = 1 \times {3^4}\]
Now, we also know that $3$ raised to power $4$ is equal to $3 \times 3 \times 3 \times 3 = 81$.

Therefore, \[{\left( { - 3} \right)^4} = 81\].

Note:These rules or laws of indices help us to minimize the problems and get the answer in very less time. These powers can be positive and negative but can be moulded according to our convenience while solving the problem. Also note that cube-root, square-root are fractions with 1 as numerator and respective root in denominator.