I^16 simplify

Raising i to the 16 power

The powers of i have a repetitive cyclic nature.
When we raise the imagenry unit i to increasing powers, we get a pattern which repeats itself.
Observe the following table of powers of i.

As you see above, the pattern repeats itself and is four members long.
Therefore, i^x+8 i^x+4 will equal^x.

When asked to determine the value of i to a power higher than 4, we can use this information in order to find our position in the cycle.
So, instead of using the actual power, we can take the remainder of the power divided by 4.

Solution for i¹⁶

16/4= 4 remainder 0
Using above knowledge
i¹⁶=
(i⁴⁾⁴×i⁰=
(1)⁴×i⁰=
1×i⁰=
i⁰=
1

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1/16 is already in the simplest form. It can be written as 0.0625 in decimal form (rounded to 6 decimal places).

Steps to simplifying fractions

1. Find the GCD (or HCF) of numerator and denominator
   GCD of 1 and 16 is 1
2. Divide both the numerator and denominator by the GCD
   1 ÷ 1/16 ÷ 1
3. Reduced fraction: 1/16
   Therefore, 1/16 simplified to lowest terms is 1/16.

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Equivalent fractions: ²/₃₂³/₄₈⁵/₈₀⁷/₁₁₂

More fractions: ²/₁₆¹/₃₂³/₁₆¹/₄₈²/₁₆¹/₁₇¹/₁₅

Calculator Use

Convert improper fractions to mixed numbers in simplest form. This calculator also simplifies proper fractions by reducing to lowest terms and showing the work involved.

In order to simplify a fraction there must be:

1. A number that will divide evenly into both the numerator and denominator so it can be reduced, or
2. The numerator must be greater than the denominator, (an improper fraction), so it can be converted to a mixed number.

What is an Improper Fraction?

An improper fraction is any fraction where the numerator is greater than the denominator. Examples of improper fractions are 16/3, 81/9, 525/71.

How to Convert an Improper Fraction to a Mixed Number

1. Divide the numerator by the denominator
2. Write down the whole number result
3. Use the remainder as the new numerator over the denominator. This is the fraction part of the mixed number.

Example: Convert the improper fraction 16/3 to a mixed number.

1. Divide 16 by 3: 16 ÷ 3 = 5 with remainder of 1
2. The whole number result is 5
3. The remainder is 1. With 1 as the numerator and 3 as the denominator, the fraction part of the mixed number is 1/3.
4. The mixed number is 5 1/3. So 16/3 = 5 1/3.

When possible this calculator first reduces an improper fraction to lowest terms before finding the mixed number form.

Example: Convert the improper fraction 45/10 to a mixed number.

• This calculator reduces the improper fraction to lowest terms by dividing numerator and denominator by the greatest common factor (GCF). The GCF of 45 and 10 is 5.

• \( \dfrac{45\div5}{10\div5} = \dfrac{9}{2}\)

• Use this reduced improper fraction and divide 9 by 2: 9 ÷ 2 = 4 with remainder of 2
• The whole number result is 4
• The remainder is 1. With 1 as the numerator and 2 as the reduced denominator, the fraction part of the mixed number is 1/2.
• The mixed number 4 1/2. So 45/10 = 4 1/2.

Related Calculators

For additional explanation of factoring numbers to find the greatest common factor (GCF) see the Greatest Common Factor Calculator.

If your improper fraction numbers are large you can use the Long Division with Remainders Calculator to find whole number and remainder values when simplifying fractions by hand.

To perform math operations on fractions before you simplify them try our Fractions Calculator. This calculator will also simplify improper fractions into mixed numbers.