Which of the following expressions is true?
1. 2^4 x 2^4 > 2^7
2. 3^2 x 3^6 = 3^7
3. 4^3 x 4^5 < 4^8
4. 5^2 x 5^3 = 5^6
Solution:
1. 2^4 x 2^4 = 16 x 16 = 256, which is greater than 2^7 = 128. So, the expression 2^4 x 2^4 > 2^7 is true.
2. 3^2 x 3^6 = 9 x 729 = 6561, which is equal to 3^7 = 2187. So, the expression 3^2 x 3^6 = 3^7 is false.
3. 4^3 x 4^5 = 64 x 1024 = 65536, which is less than 4^8 = 65536. So, the expression 4^3 x 4^5 < 4^8 is true.
4. 5^2 x 5^3 = 25 x 125 = 3125, which is equal to 5^6 = 15625. So, the expression 5^2 x 5^3 = 5^6 is false.
Therefore, the correct choice is (1) 2^4 x 2^4 > 2^7.
To determine which of the expressions is true, we can simplify each expression and compare them.
1. Using the rule of exponents (x^a x x^b = x^(a+b)), we can simplify the left side of the first expression to (2^4) x (2^4) = 2^(4+4) = 2^8. The right side is already in the form of 2^7. So, the first expression becomes 2^8 > 2^7.
2. Similarly, using the rule of exponents, we can simplify the left side of the second expression to (3^2) x (3^6) = 3^(2+6) = 3^8. The right side is already in the form of 3^7. So, the second expression becomes 3^8 = 3^7.
3. For the third expression, we can simplify the left side using the rule of exponents: (4^3) x (4^5) = 4^(3+5) = 4^8. The right side is already in the form of 4^8. So, the third expression becomes 4^8 < 4^8.
4. Lastly, we can simplify the left side of the fourth expression using the rule of exponents: (5^2) x (5^3) = 5^(2+3) = 5^5. The right side is already in the form of 5^6. So, the fourth expression becomes 5^5 = 5^6.
Based on the simplifications, we can determine that the only true expression is the second expression:
2. 3^2 x 3^6 = 3^7