the expression a^2(b^3)/a^-4(b^2) is equivalent to
1 a^6/b^5
2. b^5/a^6
3. a^2/b
4. a^-2(b^-1)
To simplify the expression a^2(b^3)/a^-4(b^2), we can use the rules of exponents.
First, let's simplify the expression in the numerator: a^2(b^3).
According to the rule of exponents, when multiplying variables with the same base, we add their exponents. So, a^2(b^3) can be written as a^2 * b^3.
Second, let's simplify the expression in the denominator: a^-4(b^2).
According to the rule of exponents, when dividing variables with the same base, we subtract their exponents. So, a^-4(b^2) can be written as a^-4 * b^2.
Now, let's simplify the entire expression by dividing the numerator by the denominator: (a^2 * b^3)/(a^-4 * b^2).
According to the rule of exponents, when dividing powers of the same base, we subtract the exponents. So, a^2/a^-4 simplifies to a^(2 - (-4)), which becomes a^6.
Similarly, b^3/b^2 simplifies to b^(3 - 2), which becomes b^1 or simply b.
Therefore, the simplified expression is a^6 * b^1, which can be written as a^6b.
So, the equivalent expression to a^2(b^3)/a^-4(b^2) is 1. a^6b.