disclaimer: official answer / explanation to the question provided below.
can anyone please explain why i cant just do log(a) on both sides of the following equation. That'll help me find b which I can use to find a. that means that i dont need EITHER (1) or (2) to solve the problem...
If a and b are nonzero integers and a^3b = a^(b – 6) , what is the value of b^a ?
(1) a^3 = 8
(2) a^2 = 4
Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
EITHER statement BY ITSELF is sufficient to answer the question.
Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, requiring more data pertaining to the problem.
The equation of the question stem is a3b = ab - 6. Let's keep in mind that when two powers have equal bases where the equal bases are not - 1, 0, or 1, then the exponents must be equal. We already know from the question stem that a is not 0. We also know that b is not 0. Now let's look at the statements.
Statement (1) says that a3 = 8. Looking at statement (1), we see that the only possible value of a is 2, since 2 3 = 8. Substituting 2 for a in the equation a3b = ab - 6, we have 2 3b = 2 b - 6. Since the equal powers 2 3b and 2 b - 6 have the same base 2, which is not one of - 1, 0, or 1, the exponents 3 b and b - 6 must be equal. So we have the equation 3 b = b - 6. This is one first-degree equation with the one variable b. We can find the single value of b from this equation. Since we have one possible value for a and one possible nonzero value for b, we will have one value for ba. Therefore, statement (1) is sufficient to find the single value for ba. We can eliminate choices (B), (C), and (E).
Statement (2) looks similar, but remember that ( - 2) 2 = (2) 2 = 4, so a could be either - 2 or 2. If we substitute 2 for a in the equation a3b = ab - 6 of the question stem, we get the equation 2 3b = 2 b - 6. Since the equal powers 2 3b and 2 b - 6 have the same base 2, which is not one of - 1, 0, or 1, the exponents 3 b and b - 6 must be equal. So we have the equation 3 b = b - 6. This is one first-degree equation with the one variable b. We can find the single value of b from this equation. Similarly, if we substitute - 2 for a in the equation a3b = ab - 6 of the question stem, we get the equation ( - 2) 3b = ( - 2) b - 6. Since the equal powers ( - 2) 3b and ( - 2) b - 6 have the same base - 2, which is not one of - 1, 0, or 1, the exponents 3 b and b - 6 must be equal. So we have the same equation 3 b = b - 6. Again, this is one first-degree equation with the one variable b. We can find the single value of b from this equation. So statement (2) says that b can have just one nonzero value. However, with two possible values for a, there are two possible values for ba. So statement (2) is insufficient. Choice (A) is correct.
can anyone please explain why i cant just do log(a) on both sides of the following equation. That'll help me find b which I can use to find a. that means that i dont need EITHER (1) or (2) to solve the problem...
If a and b are nonzero integers and a^3b = a^(b – 6) , what is the value of b^a ?
(1) a^3 = 8
(2) a^2 = 4
Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
EITHER statement BY ITSELF is sufficient to answer the question.
Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, requiring more data pertaining to the problem.
The equation of the question stem is a3b = ab - 6. Let's keep in mind that when two powers have equal bases where the equal bases are not - 1, 0, or 1, then the exponents must be equal. We already know from the question stem that a is not 0. We also know that b is not 0. Now let's look at the statements.
Statement (1) says that a3 = 8. Looking at statement (1), we see that the only possible value of a is 2, since 2 3 = 8. Substituting 2 for a in the equation a3b = ab - 6, we have 2 3b = 2 b - 6. Since the equal powers 2 3b and 2 b - 6 have the same base 2, which is not one of - 1, 0, or 1, the exponents 3 b and b - 6 must be equal. So we have the equation 3 b = b - 6. This is one first-degree equation with the one variable b. We can find the single value of b from this equation. Since we have one possible value for a and one possible nonzero value for b, we will have one value for ba. Therefore, statement (1) is sufficient to find the single value for ba. We can eliminate choices (B), (C), and (E).
Statement (2) looks similar, but remember that ( - 2) 2 = (2) 2 = 4, so a could be either - 2 or 2. If we substitute 2 for a in the equation a3b = ab - 6 of the question stem, we get the equation 2 3b = 2 b - 6. Since the equal powers 2 3b and 2 b - 6 have the same base 2, which is not one of - 1, 0, or 1, the exponents 3 b and b - 6 must be equal. So we have the equation 3 b = b - 6. This is one first-degree equation with the one variable b. We can find the single value of b from this equation. Similarly, if we substitute - 2 for a in the equation a3b = ab - 6 of the question stem, we get the equation ( - 2) 3b = ( - 2) b - 6. Since the equal powers ( - 2) 3b and ( - 2) b - 6 have the same base - 2, which is not one of - 1, 0, or 1, the exponents 3 b and b - 6 must be equal. So we have the same equation 3 b = b - 6. Again, this is one first-degree equation with the one variable b. We can find the single value of b from this equation. So statement (2) says that b can have just one nonzero value. However, with two possible values for a, there are two possible values for ba. So statement (2) is insufficient. Choice (A) is correct.
