select all the expressions that are equivalent to ((11^{-7})^{9}). (\frac{1}{11^{-63}}) (\frac{1}{11^{2}})…

Explanation:

Step1: Simplify ((11^{-7})^9)

Using the exponent rule ((a^m)^n = a^{m\times n}), we have ((11^{-7})^9 = 11^{-7\times9}=11^{-63}).

Step2: Analyze (\frac{1}{11^{-63}})

Using the rule (a^{-n}=\frac{1}{a^n}), so (\frac{1}{11^{-63}} = 11^{63})? Wait, no, wait. Wait, (a^{-n}=\frac{1}{a^n}), so (\frac{1}{a^{-n}}=a^n). Wait, but our original is (11^{-63}=\frac{1}{11^{63}}), so (\frac{1}{11^{-63}}=\frac{1}{\frac{1}{11^{63}}}=11^{63})? But also, let's check the other options. Wait, maybe I made a mistake. Wait, the original expression is ((11^{-7})^9 = 11^{-63}=\frac{1}{11^{63}}). Now let's check each option:

1. (\frac{1}{11^{-63}}): Using (a^{-n}=\frac{1}{a^n}), so (\frac{1}{11^{-63}} = 11^{63})? Wait, no, (\frac{1}{11^{-63}}=11^{63}) (since (\frac{1}{a^{-n}} = a^n)). But our original is (11^{-63}=\frac{1}{11^{63}}). Wait, maybe the first option is correct? Wait, no, let's re - evaluate.

Wait, ((11^{-7})^9 = 11^{-63}). Now, (\frac{1}{11^{-63}}=11^{63}) (because (\frac{1}{a^{-n}}=a^n)). But also, (\frac{1}{11^{63}} = 11^{-63}), so the third option (\frac{1}{11^{63}}) is equivalent. Now, what about (\frac{1}{11^{-63}}): let's see, (\frac{1}{11^{-63}}=11^{63}), but our original is (11^{-63}), so is (\frac{1}{11^{-63}}) equivalent? Wait, no, unless there's a miscalculation. Wait, no, the exponent rule: ((a^m)^n=a^{mn}), so ((11^{-7})^9 = 11^{-63}). Now, (11^{-63}=\frac{1}{11^{63}}) (by (a^{-n}=\frac{1}{a^n})). Also, (\frac{1}{11^{-63}}=11^{63}) (by (\frac{1}{a^{-n}} = a^n)), but that's not equal to (11^{-63}). Wait, maybe the first option is a typo? Wait, no, let's check again.

Wait, maybe the first option is (\frac{1}{11^{-63}}), which is (11^{63}), but our original is (11^{-63}). So that's not equivalent. Wait, the second option is (\frac{1}{11^{2}}), which is (11^{-2}), not equivalent. The fourth option is (11^{2}), not equivalent. Wait, the third option is (\frac{1}{11^{63}}), which is (11^{-63}), so that's equivalent. Wait, but the first option: (\frac{1}{11^{-63}}). Let's use the rule (a^{-n}=\frac{1}{a^n}), so (\frac{1}{11^{-63}} = 11^{63}), and (11^{-63}=\frac{1}{11^{63}}). Are (11^{63}) and (\frac{1}{11^{63}}) related? Yes, they are reciprocals. Wait, maybe I made a mistake in the exponent rule. Wait, ((11^{-7})^9=11^{-63}). Now, (\frac{1}{11^{-63}} = 11^{63}) (because (\frac{1}{a^{-n}}=a^n)), and (11^{-63}=\frac{1}{11^{63}}). So the third option (\frac{1}{11^{63}}) is equivalent. What about the first option (\frac{1}{11^{-63}}): let's see, (\frac{1}{11^{-63}} = 11^{63}), and our original is (11^{-63}). So (11^{63}) and (11^{-63}) are reciprocals, but are they equivalent? No, unless there's a mistake. Wait, maybe the first option is correct because (\frac{1}{11^{-63}}=11^{63}), but our original is (11^{-63}), so that's not. Wait, maybe the user made a mistake in the checkmarks, but let's proceed.

Wait, the correct equivalent expressions to ((11^{-7})^9) are:

First, simplify ((11^{-7})^9) using the power - of - a - power rule ((a^m)^n=a^{m\times n}). So (m=-7), (n = 9), then ((11^{-7})^9=11^{-7\times9}=11^{-63}).

Now, by the definition of negative exponents, (a^{-n}=\frac{1}{a^n}), so (11^{-63}=\frac{1}{11^{63}}). Also, (\frac{1}{11^{-63}}=11^{63}) (because (\frac{1}{a^{-n}}=a^n)). Wait, but (11^{-63}) and (11^{63}) are reciprocals. Now, let's check each option:

• Option 1: (\frac{1}{11^{-63}}). As we saw, (\frac{1}{11^{-63}} = 11^{63}). Is (11^{63}) equivalent to (11^{-63})? No, unless (63 = 0), which it's not. So this option is incorrect.

• Option 2: (\frac{1}{11^{2}}=11^{-2}), which is not equal to (11^{-63}). So this option is incorrect.

• Option 3: (\frac{1}{11^{63}}=11^{-63}), which is equal to ((11^{-7})^9). So this option is correct.

• Option 4: (11^{2}) is not equal to (11^{-63}). So this option is incorrect.

So the only correct option is the third one (\frac{1}{11^{63}}). But the original checkmarks are wrong. Let's re - do the analysis:

1. Simplify ((11^{-7})^9): Using ((a^m)^n=a^{m\times n}), we get ((11^{-7})^9 = 11^{-7\times9}=11^{-63}).

2. Analyze each option:

   • (\frac{1}{11^{-63}}): Using (\frac{1}{a^{-n}}=a^n), this is (11^{63}\neq11^{-63}).
   • (\frac{1}{11^{2}}): This is (11^{-2}\neq11^{-63}).
   • (\frac{1}{11^{63}}): Using (a^{-n}=\frac{1}{a^n}), this is (11^{-63}), which is equal to ((11^{-7})^9).
   • (11^{2}): This is not equal to (11^{-63}).

Answer:

The only expression equivalent to ((11^{-7})^9) is (\frac{1}{11^{63}}) (the third option).