Question

How do you simplify ${( - 3)^{ - 5}}$ ?

Answer 1

Hint: An exponential function is defined as a function in which a number is raised to some power, they are of the form ${a^x}$ where $a$ is the base and $x$ is its power. When a number is raised to the power “n”, it means that the number is multiplied with itself “n” times, for example, let ${a^n}$ be an exponential function, it means that the number “a” is multiplied with itself “n” times. So, ${( - 3)^{ - 5}}$means that $( - 3)$ is multiplied with itself $( - 5)$ times. There are several laws of exponents that help us to simplify the exponential functions and find their value. Using one such law, we will simplify the given exponential function and get the correct answer.

Complete step-by-step solution:
We have to simplify ${( - 3)^{ - 5}}$
We know that ${a^{ - x}} = \dfrac{1}{{{a^x}}}$
So, ${( - 3)^{ - 5}} = \dfrac{1}{{{{( - 3)}^5}}}$
${( - 3)^5}$ means $( - 3)$ multiplied with itself 5 times, so we get –
$
   \Rightarrow {( - 3)^{ - 5}} = \dfrac{1}{{ - 3 \times - 3 \times - 3 \times - 3 \times - 3}} \\
   \Rightarrow {( - 3)^{ - 5}} = \dfrac{1}{{ - 243}} \\
   \Rightarrow {( - 3)^{ - 5}} = - \dfrac{1}{{243}} \\
 $
Hence the simplified form of ${( - 3)^{ - 5}}$ is $ - \dfrac{1}{{243}}$

Note: We are given an exponential function ${( - 3)^{ - 5}}$ in this question. $( - 3)$ is raised to the power $ - 5$ , so $( - 3)$ is called the base and $ - 5$ is called its power. We have used the law of exponent which states that a number raised to some negative power is equal to the reciprocal of that number raised to the same power but with positive sign, that is, ${a^{ - x}} = \dfrac{1}{{{a^x}}}$ . The answer obtained is a fraction that is already in simplified form, else we would have to simplify the obtained fraction. This way we can find the value of any number that is raised to the power of some other number.