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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], we need to use the properties of exponents. Here we go step-by-step:

1. Identify the rule: The expression involves division of exponents with the same base. The rule we use is [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex].

2. Apply the rule: For our specific problem, we have:
[tex]\[ \frac{3^{-6}}{3^{-4}} = 3^{(-6) - (-4)} \][/tex]

3. Simplify the exponent: Subtracting [tex]\(-4\)[/tex] is the same as adding 4. So, we have:
[tex]\[ (-6) - (-4) = -6 + 4 = -2 \][/tex]

4. Write the simplified expression: With the simplified exponent, the expression now becomes:
[tex]\[ 3^{-2} \][/tex]

5. Evaluate the expression: To find the numerical value of [tex]\(3^{-2}\)[/tex], we know that a negative exponent indicates the reciprocal:
[tex]\[ 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \][/tex]

Therefore, the simplified expression [tex]\(\frac{3^{-6}}{3^{-4}}\)[/tex] results in [tex]\(3^{-2}\)[/tex], which equals [tex]\(\frac{1}{9}\)[/tex].

In summary:
[tex]\[ \frac{3^{-6}}{3^{-4}} = 3^{-2} = \frac{1}{9} \][/tex]