Q. What is \( \frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2} \)?

Answer

Compute the product as a power:
\[
\left(\tfrac{1}{2}\right)^6=\frac{1^6}{2^6}=\frac{1}{64}.
\]

Final result: \(\boxed{\tfrac{1}{64}}\).

Detailed Explanation

Write the product as a power because the same factor appears six times:

\( \left(\tfrac{1}{2}\right)^6 \)
Raise numerator and denominator separately (power of a fraction):

\( \left(\tfrac{1}{2}\right)^6 = \dfrac{1^6}{2^6} \)
Compute the powers:

\( \dfrac{1^6}{2^6} = \dfrac{1}{64} \)
Final answer:

\( \boxed{\dfrac{1}{64}} \)

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FAQs

What is the product of \( \tfrac{1}{2}\times\tfrac{1}{2}\times\tfrac{1}{2}\times\tfrac{1}{2}\times\tfrac{1}{2}\times\tfrac{1}{2} \)?

-A: \( \left(\tfrac{1}{2}\right)^6=\tfrac{1}{64}.\)

How do you compute this quickly?

-A: Use exponents: multiply six identical factors gives \( (1/2)^6 \). Or halve 1 six times.

Why is it \( (1/2)^6 \) and not something else?

-A: Repeated multiplication of the same number n times equals that number to the nth power: \(a\cdot a\cdot\ldots\cdot a = a^n\).

How do you multiply fractions in general?

-A: Multiply numerators and denominators: \( \frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd} \).

What is the decimal form?

-A: \( \tfrac{1}{64}=0.015625.\)

What is the percent?

-A: \(0.015625=1.5625\%\) (multiply decimal by 100).

How is this written using powers of 2?

-A: \( \left(\tfrac{1}{2}\right)^6 = 2^{-6} = \tfrac{1}{2^6}.\)

What is the reciprocal?

-A: The reciprocal is \(64\) because \( \frac{1}{1/64}=64.\)

-Any probability interpretation?

-A: Probability of getting six heads in six fair coin flips is \( (1/2)^6=\tfrac{1}{64}.\)

Show the stepwise halving.

-A: Start at 1: \(1\to\tfrac12\to\tfrac14\to\tfrac18\to\tfrac1{16}\to\tfrac1{32}\to\tfrac1{64}.\)

Common mistakes students make?

-A: -Adding denominators instead of multiplying, forgetting exponents, or simplifying incorrectly. For identical fractions use powers: \( (a/b)^n=\tfrac{a^n}{b^n}\).

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