The result of multiplying 2/3 by itself three times is 8/27.

When calculating 2/3 times 2/3 times 2/3 in fraction form, the process involves multiplying all the numerators together to find the new numerator and multiplying all the denominators together to find the new denominator. This specific operation represents the "cube" of the fraction 2/3, which is written mathematically as $(2/3)^3$.

Step by Step Calculation of 2/3 Times 2/3 Times 2/3

To solve this problem accurately, it is best to break it down into two distinct stages of multiplication or use the simultaneous multiplication rule for fractions.

Stage 1: The Numerators

The numerator is the top number of a fraction, representing the parts being counted. In this case, the numerator is 2.

  1. Multiply the first two numerators: $2 \times 2 = 4$.
  2. Multiply the result by the third numerator: $4 \times 2 = 8$. The final numerator for the product is 8.

Stage 2: The Denominators

The denominator is the bottom number, representing the total number of parts that make up a whole. Here, the denominator is 3.

  1. Multiply the first two denominators: $3 \times 3 = 9$.
  2. Multiply the result by the third denominator: $9 \times 3 = 27$. The final denominator for the product is 27.

Combining the Results

By placing the new numerator over the new denominator, the resulting fraction is 8/27.

Understanding the Concept of Cubing a Fraction

In mathematics, when a number or a fraction is multiplied by itself three times, it is referred to as being "cubed" or raised to the third power. This is a common operation in algebra and geometry.

The general rule for raising a fraction to a power is expressed as: $$(a/b)^n = a^n / b^n$$

Applying this rule to $2/3$: $$(2/3)^3 = 2^3 / 3^3 = (2 \times 2 \times 2) / (3 \times 3 \times 3) = 8/27$$

This rule highlights that you are essentially performing two separate exponentiation tasks—one for the top and one for the bottom. Based on years of observing students tackle these problems, I have found that those who visualize the exponent applying to both the numerator and the denominator separately tend to make fewer mistakes than those who try to solve the entire fraction at once.

Why 8/27 is the Simplest Form

After arriving at a fraction, the next logical step is to determine if it can be reduced. A fraction is in its simplest form (or lowest terms) when the numerator and the denominator share no common factors other than 1. These numbers are called "coprime" or "relatively prime."

To verify if 8/27 can be simplified, we examine the prime factors of both numbers:

  • Prime factors of 8: $2 \times 2 \times 2$ (or $2^3$)
  • Prime factors of 27: $3 \times 3 \times 3$ (or $3^3$)

As we can see, the factors of 8 are entirely composed of the number 2, while the factors of 27 are entirely composed of the number 3. Since there are no overlapping prime factors between 8 and 27, the greatest common divisor (GCD) is 1. Therefore, 8/27 is already in its simplest form and cannot be reduced further.

Visualization Through Geometry

One of the most effective ways to internalize why $2/3 \times 2/3 \times 2/3$ equals 8/27 is to imagine a physical cube.

Imagine a cube where each side has a length of 2/3 of a unit. To find the volume of this cube, you must multiply length by width by height ($V = s^3$).

  • If the cube had a side length of 1 unit, the volume would be $1 \times 1 \times 1 = 1$ cubic unit.
  • With a side length of 2/3, you are essentially taking two-thirds of the length, two-thirds of the width, and two-thirds of the height.

If you were to divide a standard $1 \times 1 \times 1$ cube into smaller sub-cubes, each with a side of 1/3, you would have a total of 27 small cubes (since $3 \times 3 \times 3 = 27$). Your specific cube with a side length of 2/3 would occupy exactly 8 of those 27 sub-cubes. This visual representation makes it clear why the volume is 8 out of 27 parts of the whole.

The Role of This Calculation in Probability

In the realm of probability, multiplying fractions is a daily necessity. This specific calculation—$2/3 \times 2/3 \times 2/3$—is used to determine the likelihood of an event with a 2/3 chance of occurring happening three times in a row, assuming each event is independent.

For example, if you have a jar where 2 out of every 3 marbles are blue, and you draw a marble, record its color, and put it back three times, the probability of drawing three blue marbles in a row is: $$(2/3) \times (2/3) \times (2/3) = 8/27$$

In percentage terms, 8/27 is approximately 29.6%. In professional statistical analysis, maintaining the fraction form (8/27) is often preferred because it is an exact value, whereas the decimal 0.296 is an approximation.

Comparison Between Multiplication and Addition of Fractions

A frequent point of confusion for those re-learning math is the difference between multiplying a fraction three times and adding it three times.

  • Multiplication: $2/3 \times 2/3 \times 2/3 = 8/27$. This represents finding "two-thirds of two-thirds of two-thirds." Multiplication of fractions between 0 and 1 always results in a smaller value.
  • Addition: $2/3 + 2/3 + 2/3 = 6/3 = 2$. This represents taking two-thirds and repeating it three times. Addition of positive fractions always results in a larger value.

In my experience, students who accidentally add instead of multiply often realize their error when they see the result is larger than the original fraction. If you are calculating a part of a part, the number should logically get smaller. 8/27 (which is roughly 0.296) is significantly smaller than 2/3 (which is roughly 0.666).

Converting 8/27 to Decimal Form

While the query specifically asks for the "fraction form," it is often helpful for comparative purposes to know the decimal equivalent. To convert 8/27 to a decimal, you perform long division: 8 divided by 27.

  1. 27 goes into 80 two times ($27 \times 2 = 54$), leaving a remainder of 26.
  2. 27 goes into 260 nine times ($27 \times 9 = 243$), leaving a remainder of 17.
  3. 27 goes into 170 six times ($27 \times 6 = 162$), leaving a remainder of 8.

The pattern begins to repeat at this point. Thus, the decimal form is $0.296296...$, which can be written as $0.\overline{296}$. This repeating decimal is a characteristic of fractions whose denominators have prime factors other than 2 or 5.

Common Pitfalls to Avoid

When working with repeated fraction multiplication, keep these three common mistakes in mind:

  1. Only multiplying the numerators: Some mistakenly calculate $(2 \times 2 \times 2) / 3$, resulting in 8/3. This is incorrect because the scaling applies to the entire ratio, not just the top part.
  2. Finding a common denominator: Unlike addition, you do not need a common denominator for multiplication. Some waste time trying to convert the fractions before multiplying, which is unnecessary when the denominators are already the same.
  3. Incorrect reduction: It is tempting to try and "divide" the final result by 2 or 3, but as established earlier, 8 and 27 have no common factors. Always check the prime factorization if you are unsure.

Summary of the Result

The operation $2/3 \times 2/3 \times 2/3$ yields the fraction 8/27. This is achieved by multiplying the three numerators (8) and the three denominators (27). The fraction is in its simplest form because 8 and 27 share no common divisors. This calculation is mathematically equivalent to "cubing" the fraction and is a foundational concept in geometry, probability, and algebra.

Frequently Asked Questions

How do I type 2/3 times 2/3 times 2/3 into a scientific calculator?

On most scientific calculators, you can use the parentheses and the exponent key. You would type ( 2 / 3 ) ^ 3 or ( 2 / 3 ) x^y 3. If you do not have an exponent key, simply type ( 2 / 3 ) * ( 2 / 3 ) * ( 2 / 3 ).

Is 8/27 a terminating or repeating decimal?

It is a repeating decimal ($0.\overline{296}$). This is because the denominator, 27, is $3^3$. Only fractions with denominators whose prime factors are strictly 2 or 5 (like 1/2, 1/4, 1/5, 1/8, 1/10) will result in a terminating decimal.

Can this be represented as a mixed number?

No, 8/27 is a "proper fraction" because the numerator is smaller than the denominator. Mixed numbers are only used for "improper fractions" where the numerator is larger than or equal to the denominator (e.g., 27/8 would be $3 \frac{3}{8}$).

What is 2/3 times 2/3 times 2/3 as a percentage?

To convert the fraction to a percentage, multiply the decimal $0.296296...$ by 100. This gives approximately 29.63%.

What happens if the exponent is negative, such as (2/3)^-3?

If the exponent were negative, you would take the reciprocal of the fraction and then cube it. So, $(2/3)^{-3}$ becomes $(3/2)^3$, which is $27/8$.