Question

1 ( 3^4 cdot 3^5 = 3^9 ) true false
2 ( (8^7)^3 = 8^{10} ) true false
3 ( \frac{4^8}{4^2} = 4^3 ) true false
4 ( \frac{9^5}{9^7} = 9^1 ) true false
5 ( 5^3 cdot 5^2 = 5^6 ) true false
6 ( 1 = 2^0 ) true false
7 ( (2^4)^5 = 2^9 ) true false
8 ( 5^{-1} = \frac{1}{5^1} ) true false
9 ( 9^4 cdot 9^{-6} = 9^1 ) true false
10 ( \frac{6^{12}}{6^{12}} = 6^0 ) true false
11 ( (7^3)^{-5} cdot 7^{10} = 7^{25} ) true false
12 ( \frac{1}{2^6 cdot 2^8} = 2^{-14} ) true false
13 ( \frac{(4^5)^3}{4^8} = 4^7 )
14 ( \frac{12^0}{12^3} = \frac{1}{12^3} )
15 ( \frac{4^5 cdot 4^{10}}{4^2} = 4^{13} ) true f

Explanation:

Step1: Recall Exponent Rules

We use exponent rules: \(a^m \cdot a^n = a^{m + n}\), \((a^m)^n = a^{mn}\), \(\frac{a^m}{a^n}=a^{m - n}\), \(a^0 = 1\), \(a^{-n}=\frac{1}{a^n}\)

Step2: Analyze Each Problem

1. \(3^4\cdot3^5\): By \(a^m\cdot a^n=a^{m + n}\), \(3^{4 + 5}=3^9

eq3^2\). So "False" was wrong, correct is "False"? Wait, original marked "True" which is wrong.

1. \((8^7)^3\): By \((a^m)^n=a^{mn}\), \(8^{7\times3}=8^{21}

eq8^9\). So marked "False" is correct.

1. \(\frac{4^9}{4^2}\): By \(\frac{a^m}{a^n}=a^{m - n}\), \(4^{9 - 2}=4^7\). So marked "False" is wrong, correct is "True".
2. \(\frac{9^7}{9^7}\): \(9^{7-7}=9^0 = 1

eq9^1\). So "True" is wrong, correct "False".

1. \(5^3\cdot5^2\): \(5^{3 + 2}=5^5\). Marked "False" is wrong, correct "True".
2. \(1 = 2^0\), \(2^3=8

eq1\). So "False" is correct.

1. \((2^4)^5\): \(2^{4\times5}=2^{20}

eq2^9\). So "True" is wrong, correct "False".

1. \(5^{-1}=\frac{1}{5^1}=\frac{1}{5}\). So "True" is correct.
2. \(9^4\cdot9^{-6}\): \(9^{4-6}=9^{-2}=\frac{1}{9^2}

eq9^1\). So "False" is correct.

1. \(\frac{6^{12}}{6^{12}}\): \(6^{12 - 12}=6^0 = 1\). So "True" is correct.
2. \((7^3)^{-5}\cdot7^{10}\): \((7^3)^{-5}=7^{-15}\), \(7^{-15}\cdot7^{10}=7^{-15 + 10}=7^{-5}

eq7^{25}\). So "False" is correct.

1. \(\frac{1}{2^6\cdot2^8}\): \(2^6\cdot2^8=2^{14}\), \(\frac{1}{2^{14}}=2^{-14}\). So "True" is correct.
2. \(\frac{(4^5)^2}{4^8}\): \((4^5)^2 = 4^{10}\), \(\frac{4^{10}}{4^8}=4^{10 - 8}=4^2\). So "True" is correct.
3. \(\frac{12^0}{12^3}\): \(12^0 = 1\), \(\frac{1}{12^3}\). So "True" is correct.
4. \(\frac{4^4\cdot4^{10}}{4^2}\): \(4^{4 + 10-2}=4^{12}

eq4^{16}\). So "False" is correct.

Answer:

For each problem, the correct True/False (correcting the marked ones):

1. False (marked True incorrectly)
2. False (marked correctly)
3. True (marked False incorrectly)
4. False (marked True incorrectly)
5. True (marked False incorrectly)
6. False (marked correctly)
7. False (marked True incorrectly)
8. True (marked correctly)
9. False (marked correctly)
10. True (marked correctly)
11. False (marked correctly)
12. True (marked correctly)
13. True (marked correctly)
14. True (marked correctly)
15. False (marked True incorrectly)