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Describe how to simplify the expression [tex]\frac{3^{-6}}{3^{-4}}[/tex].

A. Divide the bases and then add the exponents.
B. Keep the base the same and then add the exponents.
C. Multiply the bases and then subtract the exponents.
D. Keep the base the same and then subtract the exponents.

Sagot :

To simplify the expression [tex]\(\frac{3^{-6}}{3^{-4}}\)[/tex], you follow these steps:

1. Identify the Operation on Exponents:
- The base is the same in both parts of the fraction, which is [tex]\(3\)[/tex].

2. Subtract the Exponents:
- When dividing powers of the same base, you subtract the exponent of the denominator from the exponent of the numerator.

[tex]\[ \frac{a^m}{a^n} = a^{m-n} \][/tex]

3. Apply the Rule:
- Here, the exponents are [tex]\(-6\)[/tex] and [tex]\(-4\)[/tex]. So, you subtract [tex]\(-4\)[/tex] from [tex]\(-6\)[/tex]:

[tex]\[ -6 - (-4) = -6 + 4 = -2 \][/tex]

4. Simplify the Expression:
- Therefore, the given expression simplifies to:

[tex]\[ \frac{3^{-6}}{3^{-4}} = 3^{-2} \][/tex]

5. Optional - Convert the Negative Exponent to Positive:
- To express [tex]\(3^{-2}\)[/tex] in a more conventional form, recognize that a negative exponent indicates the reciprocal:

[tex]\[ 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \][/tex]

Thus, the simplified expression is [tex]\(3^{-2}\)[/tex], which is equivalent to [tex]\(\frac{1}{9}\)[/tex] or approximately [tex]\(0.1111111111111111\)[/tex].
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