91Ó°ÊÓ

The square root is a fundamental concept in mathematics. Finding the square root of a number means identifying a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because

• 4 multiplied by 4 results in 16.

When dealing with expressions like \(\sqrt{a \times b}\), the property of square roots allows us to split it as \(\sqrt{a} \times \sqrt{b}\). This property is particularly useful when simplifying scientific notation calculations.
The given expression \(\sqrt{0.00000004}\) in scientific notation as \(\sqrt{4 \times 10^{-8}}\) can be split into two parts, \(\sqrt{4}\) and \(\sqrt{10^{-8}}\). This way, we deal with smaller, simpler parts - first, an integer, and second, a power of ten.

Exponents are used to express repeated multiplication of a number by itself. For instance, \(2^3\) means \(2 \times 2 \times 2\). In scientific notation, exponents help simplify very large or small numbers. An exponent indicates how many times to multiply the base by itself.
In the expression \(10^{-8}\), -8 is the exponent and tells us how many times to divide 1 by 10, resulting in a very small number.
Applying the square root to \(10^{-8}\) involves halving this exponent, -8 becomes -4, which reflects the rule:

• \((x^m)^{1/2} = x^{m/2}\)

This helps us transform \(\sqrt{10^{-8}}\) into \(10^{-4}\). Understanding exponents is crucial when working with scientific notation, particularly in simplifying expressions and solving problems involving roots and powers.

Scientific notation is a shorthand way of expressing very large or small numbers, making them easier to read and work with. A number is expressed as a product of a coefficient (a number typically between 1 and 10) and 10 raised to an exponent.
To convert a decimal number to scientific notation, identify how many places you move the decimal point to form a new number between 1 and 10. For the decimal \(0.00000004\), you would move the decimal 8 places to the right, achieving a number \(4\). This movement directs you to use \(-8\) as the exponent in \(10^{-8}\).
The final form becomes \(4 \times 10^{-8}\).

• Coefficient is 4.
• Exponent reflects the shift: -8.

This conversion to scientific notation simplifies how we perform and understand complex operations, like finding square roots. It decomposes the complicated decimal into simpler components, ready for mathematical operations.