Solved on Dec 23, 2023

Simplify the expression 1256101\frac{1}{256^{101}} as a whole number with a negative exponent.

STEP 1

Assumptions
1. We have the expression 1256101\frac{1}{256^{101}}.
2. We want to rewrite the expression as a whole number with a negative exponent.

STEP 2

Recognize that 256256 is a power of 22, specifically 256=28256 = 2^8.

STEP 3

Rewrite the denominator using the power of 22.
256101=(28)101256^{101} = (2^8)^{101}

STEP 4

Apply the power of a power rule (ab)c=abc(a^b)^c = a^{bc} to simplify the expression.
(28)101=28×101(2^8)^{101} = 2^{8 \times 101}

STEP 5

Multiply the exponents to simplify further.
28×101=28082^{8 \times 101} = 2^{808}

STEP 6

Now we can rewrite the original expression using the simplified denominator.
1256101=12808\frac{1}{256^{101}} = \frac{1}{2^{808}}

STEP 7

Recognize that 1an=an\frac{1}{a^n} = a^{-n}.

STEP 8

Apply this property to rewrite the expression as a whole number with a negative exponent.
12808=2808\frac{1}{2^{808}} = 2^{-808}
The expression 1256101\frac{1}{256^{101}} written as a whole number with a negative exponent is 28082^{-808}.

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