Question

What is the 5th root of $160$?

Answer 1

Hint: Here, in the given question, we need to find the 5th root of $160$. 5th root of a number is the number which when multiplied by itself five times gives the original number. The 5th root of $160$ can be written as $\sqrt[5]{{160}}$. We will first find the factors of $160$ in exponential form. After this, we will find the 5th root of the factors and find the solution for the 5th root of $160$.

Complete answer:
Factors of $160$ in exponential form can be given as,
$ \Rightarrow 160 = 2 \times 2 \times 2 \times 2 \times 2 \times 5$
Now we will take 5th root of the above equation, we get
$ \Rightarrow \sqrt[5]{{160}} = \sqrt[5]{{2 \times 2 \times 2 \times 2 \times 2 \times 5}}$
It can also be written as,
$ \Rightarrow \sqrt[5]{{160}} = \sqrt[5]{{{2^5} \times 5}}$
$ \Rightarrow \sqrt[5]{{160}} = \sqrt[5]{{{2^5}}} \times \sqrt[5]{5}$
We know that $\sqrt[5]{{{2^5}}} = 2$. Putting this value in the above equation, we get,
$ \Rightarrow \sqrt[5]{{160}} = 2 \times \sqrt[5]{5}$
The value of $\sqrt[5]{5}$ is $1.37972966$, putting this value in the above equation, we get
$ \Rightarrow \sqrt[5]{{160}} = 2 \times 1.37972966$
On multiplication, we get
$ \Rightarrow \sqrt[5]{{160}} = 2.75945932$
We can also write it as,
$ \Rightarrow \sqrt[5]{{160}} = 2.7594$
Therefore, the 5th root of $160$ is $2.7594$.

Additional information: Remember that 5th root of $160$ can be represented as ${\left( {160} \right)^{\dfrac{1}{5}}}$ and $\sqrt[5]{{160}}$. In $\sqrt[5]{{160}}$, the symbol $\sqrt {} $, is called the radical sign, $160$ is the radicand (it is the number below the radical sign), and $5$ is the index.

Note: Students should do the proper factorization of the given number. Students should know the 5^th root of basic small numbers. Remember that writing four numbers after the decimal point is enough. Students are not supposed to write all the numbers after the decimal point.
Below is a tabular form for the 5^th root of numbers from $1$ to $10$.

Number	Cube root $\sqrt[5]{n}$
$1$	$1.000$
$2$	$1.1487$
$3$	$1.2457$
$4$	$1.3195$
$5$	$1.3797$
$6$	$1.4310$
$7$	$1.4758$
$8$	$1.5157$
$9$	$1.5518$
$10$	$1.5849$