How to Calculate the Square Root of a Power of 10, for Example the Square Root of 10^-31 or 10^-15.
The Correct Answer and Explanation is:
To calculate the square root of a power of 10, such as √(10⁻³¹) or √(10⁻¹⁵), follow these steps:
General Rule:
The square root of 10 raised to a power is equal to 10 raised to half of that power. 10n=10n/2\sqrt{10^n} = 10^{n/2}10n=10n/2
Example 1: √(10⁻³¹)
Apply the rule: 10−31=10−31/2=10−15.5\sqrt{10^{-31}} = 10^{-31/2} = 10^{-15.5}10−31=10−31/2=10−15.5
So, the square root of 10⁻³¹ is 10⁻¹⁵.⁵.
Example 2: √(10⁻¹⁵)
10−15=10−15/2=10−7.5\sqrt{10^{-15}} = 10^{-15/2} = 10^{-7.5}10−15=10−15/2=10−7.5
So, the square root of 10⁻¹⁵ is 10⁻⁷.⁵.
Explanation:
When dealing with powers of 10, remember that square roots and exponents are related. Taking the square root of a number is the same as raising that number to the one-half power. So when you have: 10n=(10n)1/2\sqrt{10^n} = (10^n)^{1/2}10n=(10n)1/2
By the laws of exponents, you multiply the exponents: (10n)1/2=10n×1/2=10n/2(10^n)^{1/2} = 10^{n \times 1/2} = 10^{n/2}(10n)1/2=10n×1/2=10n/2
This rule works regardless of whether the exponent is positive, negative, or a decimal. In scientific and engineering problems, exponents like 10⁻³¹ or 10⁻¹⁵ often represent very small values, such as atomic scales or extremely short time intervals. Taking their square root allows scientists to simplify equations or convert between units.
For example, if a quantity is proportional to 10⁻³¹, and a formula involves the square root of that quantity, the answer becomes 10⁻¹⁵.⁵. Knowing this exponent rule saves time and helps avoid the use of a calculator. It is especially useful in logarithmic scales, scientific notation, and exponential equations.
