Question

How do you write \[{10^{ - 4}}\] as both a decimal and a fraction?

Answer 1

Hint: We will use the laws of exponents to convert \[{10^{ - 4}}\] into a fraction. Using the fractional form, we will find the decimal equivalent of \[{10^{ - 4}}\] by dividing the numerator by the denominator. A decimal number is a number that consists of a whole number part and fractional part separated by a point called a decimal point.

Formula used:
Law of exponent: ${a^{ - m}} = \dfrac{1}{{{a^m}}};m > 0$

Complete step by step solution:
We have to convert \[{10^{ - 4}}\] into a decimal and a fraction. Let us first convert it into a fraction.
To convert \[{10^{ - 4}}\] into a fractional form, we will use the law of exponent ${a^{ - m}} = \dfrac{1}{{{a^m}}};m > 0$.
Here, $a = 10$ and $m = 4$ and $m > 0$.
So, we have
${10^{ - 4}} = \dfrac{1}{{{{10}^4}}}$.
We know that raising any number to an exponent means to multiply that number by itself the same number of times as in the exponent.
Thus, ${10^4}$ means that we have to multiply 10 by itself 4 times.
$\therefore {10^4} = 10 \times 10 \times 10 \times 10$
Hence, we can write \[{10^{ - 4}}\] in fractional form as
${10^{ - 4}} = \dfrac{1}{{10 \times 10 \times 10 \times 10}} = \dfrac{1}{{10000}}$
Now, we have to convert \[{10^{ - 4}}\] in the decimal form.
Let us perform division for $\dfrac{1}{{10000}}$ and find the decimal expansion.
Since 1 is not divisible by 10000, we put a zero in the quotient followed by a decimal point. Subtracting 0 from 1, we get the remainder as 1. Since we have put a decimal point in the quotient, we will affix one zero to the remainder.
Again, 10 is not divisible by 10000, so we put a zero in the quotient. We repeat this procedure 3 times until we get a 10000 in the dividend. When we do, we will multiply 10000 by 1 to get 10000, and finally, we get the remainder as 0.

Therefore, the decimal expansion of \[{10^{ - 4}}\] is \[{10^{ - 4}} = 0.0001\].

Note:
To divide a whole number by a power of 10 is relatively easy. We usually write the whole number and then count the number of zeroes in the power of 10. We then place a decimal point to the right of the whole number starting at the unit’s digit counting as many spaces as the number of zeroes. If there are no digits in the spaces after the decimal point, we will fill those spaces with zeros.