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Solved on Dec 23, 2023

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

STEP 1

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

STEP 2

Recognize that $256$ is a power of $2$ , specifically $256 = 2^8$ .

STEP 3

Rewrite the denominator using the power of $2$ .
$256^{101} = (2^8)^{101}$

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

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