Q. \[ \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \] in fraction form.

Q: What is \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} in fraction form?

Multiply numerators and denominators: \frac{1 \cdot 1 \cdot 1 \cdot 1}{3 \cdot 3 \cdot 3 \cdot 3} = \frac{1}{81}.

Q: How do you write that product using exponents?

Use power notation: \left(\frac{1}{3}\right)^4 = \frac{1^4}{3^4} = \frac{1}{3^4} = \frac{1}{81}.

Q: Can \frac{1}{81} be simplified further?

No. \frac{1}{81} is already in lowest terms because the numerator 1 has no common factors with 81.

Q: What is \frac{1}{81} as a decimal?

\frac{1}{81} \approx 0.012345679012345679\dots; the decimal repeats the 9-digit cycle 012345679.

Q: What is the reciprocal of \frac{1}{81}?

The reciprocal is \frac{81}{1} = 81.

Answer

Multiply numerators and denominators: \( \frac{1}{3}\times\frac{1}{3}\times\frac{1}{3}\times\frac{1}{3}=\frac{1}{3^4}=\frac{1}{81} \).

Final result: \( \frac{1}{81} \)

Detailed Explanation

Write the multiplication to be evaluated:

\( \displaystyle \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \)
Use the rule for multiplying fractions: to multiply fractions, multiply the numerators together and multiply the denominators together. In general,

\( \displaystyle \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \)
Apply the rule to the first two fractions (multiply numerators and denominators):

\( \displaystyle \frac{1}{3} \times \frac{1}{3} = \frac{1 \times 1}{3 \times 3} = \frac{1}{9} \)
Multiply the result by the third fraction:

\( \displaystyle \frac{1}{9} \times \frac{1}{3} = \frac{1 \times 1}{9 \times 3} = \frac{1}{27} \)
Multiply that result by the fourth fraction:

\( \displaystyle \frac{1}{27} \times \frac{1}{3} = \frac{1 \times 1}{27 \times 3} = \frac{1}{81} \)
Alternatively, recognize repeated multiplication of the same fraction as an exponent:

\( \displaystyle \left(\frac{1}{3}\right)^4 = \frac{1^4}{3^4} = \frac{1}{81} \)
Final answer (fraction form):

\( \displaystyle \frac{1}{81} \)

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FAQs

What is \(\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3}\) in fraction form?

Multiply numerators and denominators: \(\frac{1 \cdot 1 \cdot 1 \cdot 1}{3 \cdot 3 \cdot 3 \cdot 3} = \frac{1}{81}\).

How do you write that product using exponents?

Use power notation: \(\left(\frac{1}{3}\right)^4 = \frac{1^4}{3^4} = \frac{1}{3^4} = \frac{1}{81}\).

Can \(\frac{1}{81}\) be simplified further?

No. \(\frac{1}{81}\) is already in lowest terms because the numerator 1 has no common factors with 81.

What is \(\frac{1}{81}\) as a decimal?

\(\frac{1}{81} \approx 0.012345679012345679\dots\); the decimal repeats the 9-digit cycle 012345679.

What is the reciprocal of \(\frac{1}{81}\)?

The reciprocal is \(\frac{81}{1} = 81\).

Why do we multiply numerators and denominators when multiplying fractions?

Could you cancel before multiplying in this problem?

No, because all numerators are 1 and share no factors with 3. For other fractions, cancel common factors between any numerator and any denominator first to simplify.

What is a real-world interpretation of \(\left(\frac{1}{3}\right)^4\)?

It can represent the probability that four independent events each with probability \(\frac{1}{3}\) all occur: overall probability \(= \left(\frac{1}{3}\right)^4 = \frac{1}{81}\).

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