Find the value of $\sqrt[3]{\sqrt{0.015625}}$ is:

Square-Square Root and Cube-Cube Root

Square: If $n$ is a number then it’s square is represented by $n$ raised to the power $2$, i.e., $n^2$ and its square root is expressed as $\sqrt{n}$' where √’ is called radical. The value under the root symbol is said to be radicand.

Note:

1. $1^2$ equal to $1$

2. Square of positive numbers are always positive in nature.

3. The square of negative numbers is also positive in nature. For example, ${\left(-3\right)}^2=9$

4. Square of zero is zero

5. The square of the root of a number is equal to the value under the root. For example, ${\left(\sqrt{3}\right)}^2=3$

6. The unit place of square of any even number will have an even number only.

7. If a number has $1$ or $9$ in the unit’s place, then its square ends in $1$.

8. If a number has $4$ or $6$ in the unit’s place, then its square ends in $6$.

9. A number having $2, 3, 7$ or $8$ at unit’s place is never a perfect square. In other words, no square number ends in $2, 3, 7$ or $8$.

10. For any natural number m greater than $1$, $\left(2m,m^2-1,m^2+1\right)$ is a Pythagorean triplet.

11. Squares of even numbers are always even numbers and square of odd numbers are always odd.

12. The Square of a natural number other than one is either a multiple of $3$ or exceeds a multiple of $3$ by $1$. In other words, a perfect square leaves the remainder $0$ or $1$ on division by $3$.

Procedure to check whether a given natural number is a perfect square or not.

Step I: Obtain the natural number.

Step II: Write the number as a product of prime factors.

Step III: Group the factors in pairs in such a way that both the factors in each pair are equal.

Step IV: See whether some factor is left over or not. If no factor is left over in the grouping, then the given number is a perfect square. Otherwise, it is not a perfect-square.

Step V: To obtain the number whose square is the given number taken over one factor from each group and multiply them.

Square Roots: The square root of a number a is that number which when multiplied by itself gives a as the product. Thus, if b is the square root of a number a, then $b\times b=a$ or $b^2=a$. The square root symbol is $\sqrt{a}$.  It follows from this that $b=\sqrt{a}\iff b^2=a$

i.e. b is the square root of a if and only if a is the square of b.

Properties of Square Roots

Property 1: If the units digit of a number is 2, 3, 7 or 8, then it does not have root in N (the set of natural numbers).

Property 2: If a number ends in an odd number of zeros, then it does not have a square root. If a square number is followed by an even number of zeros, it has a square root in which the number of zeros in the end is half the number of zeros in the number.

Property 3: The square root of an even square number is even and that root of an odd square number is odd.

Property 4: If a number has a square root in N, then its unit digit must be 0, 1, 4, 5, or 9.

Property 5: Negative numbers have no square root in the system of rational numbers.

$\sqrt{\left(-9\right)}$ is not a rational number. It will be a complex number.

Property 6: The sum of first $n$ odd numbers is $n^2$.

Square Root by Prime Factorization Method

Step I: Obtain the given number.

Step II: Resolve the given number into prime factors by successive division.

Step III: Make pairs of prime factors such that both the factors in each pair are equal. Since the number is a perfect square, you will be able to make an exact number of pairs of prime factors.

Step IV: Take one factor from each pair.

Step V: Find the product of factors obtained in step IV.

Step VI: The product obtained in step V is the required square root.

Square Root of Rational Numbers in the Form of Fractions:

Step I: Obtain the fraction

Step II: If the given square root of the numerator and the denominator are the square roots of numerator and denominator respectively of the given fraction.

Step III: Find the square root of the numerator and denominator separately.

Step IV: Obtain the fraction whose numerator and denominator are the square roots of numerator and denominator respectively of the given fraction.

Step V: The fraction obtained in Step IV is the square root of the given fraction.

Cube of Numbers: The number with exponent 3 is called cube.

The word ‘cube’ is used in geometry. A cube is a solid figure which has all its sides equal.

There are only ten perfect cubes from 1 to 1000.

If a number is a number, then the cube of a is $a^3$

$a\times a\times a=a^3$

Some Properties of Cubes of Natural Numbers are:-

Property 1: Cubes of all even numbers are even.

Property 2: Cubes of all odd natural numbers are odd.

Property 3: The sum of the cubes of first n natural numbers is equal to the square of their sum.

Property 4: Cubes of the numbers ending in digit 1, 4, 5, 6 and 9 are the numbers ending in the same digit.

Cube Root: A number $m$ is the cube root of a number $n$, if $n=m^3$.

In other words, the cube-root of a number n is that number whose cube gives $n$.

The cube-root of a number is denoted by $\sqrt[3]{n}$. $\sqrt[3]{n}$ is also called a radical, $n$ is called the radicand and $3$ is called the index of the radical.

Here; $27=3^3$ $\therefore \sqrt[3]{27}=3$

Cube Root by Prime Factorization

Step I: Obtain the given number.

Step II: Resolve it into prime factors.

Step III: Group the factors in $3$ in such a way that each number of the group is same.

Step IV: Take one factor from each group.

Step V: Find the product of the factors obtained in step IV. This product is the required cube root.

Cube-root of rational-numbers: $\sqrt[3]{\left(\frac{a}{b}\right)}=\left(\frac{\sqrt[3]{a}}{\sqrt[3]{b}}\right)$