Question

What is $-\dfrac{2}{3}$ raised to the ${{5}^{th}}$ power?

Answer 1

Hint: The problem is all about indices and powers. This requires the knowledge of the properties of indices, especially the property ${{\left( ab \right)}^{m}}$ can be written as ${{a}^{m}}{{b}^{m}}$ . This is used to write $-\dfrac{2}{3}$ as $-1\times \dfrac{2}{3}$ . Then, we raise them to the powers.

Complete step by step answer:
Power or indices or exponents are a way to represent multiple or repetitive multiplications. For example, the expression $2\times 2\times 2\times 2\times 2$ can be expressed as ${{2}^{5}}$ where $2$ is called the base number and $5$ is called the exponent or power or index. It is basically a shortcut to avoid long iterative and tedious multiplications. There are various properties and laws associated with the indices.
The given base number that we have in the given problem is $-\dfrac{2}{3}$ . We are required to find the number which is $-\dfrac{2}{3}$ raised to the ${{5}^{th}}$ power. In other words, we need to find ${{\left( -\dfrac{2}{3} \right)}^{5}}$ . Now, $-\dfrac{2}{3}$ can be written as $-1\times \dfrac{2}{3}$ . The power then becomes,
$\Rightarrow {{\left( -1\times \dfrac{2}{3} \right)}^{5}}$
There is a property associated with indices. It says that if there is an expression ${{\left( ab \right)}^{m}}$ , then it can be written as ${{a}^{m}}{{b}^{m}}$ . Similarly, the expression ${{\left( -1\times \dfrac{2}{3} \right)}^{5}}$ can be written as ${{\left( -1 \right)}^{5}}\times {{\left( \dfrac{2}{3} \right)}^{5}}$ . Now, ${{\left( -1 \right)}^{5}}$ means nothing but $-1$ . And, \[{{\left( \dfrac{2}{3} \right)}^{5}}\] can be written as \[\dfrac{{{2}^{5}}}{{{3}^{5}}}\] . \[\dfrac{{{2}^{5}}}{{{3}^{5}}}\] is $\dfrac{32}{243}$ . So, the product of the two becomes $-1\times \dfrac{32}{243}=-\dfrac{32}{243}$ .

Thus, we can conclude that $-\dfrac{2}{3}$ raised to the ${{5}^{th}}$ power is $-\dfrac{32}{243}$ .

Note: We must be careful to observe the negative as that’s what the problem is all about. Also, we should know some of the basic properties of indices, else we cannot solve this problem. We should have patience while repetitive multiplication and should avoid making errors.