Question

The expression (a^3b^-4/2a^-1)^-2 represents how much Patrick won from Michael. Simplify and write your answer with positive exponents. Assume all variables are nonzero.

Answer 1

The simplified form of the expression (a^3b^{-4}/2a^{-1})^{-2} is 4a^4b^8, which is achieved by taking the reciprocal due to the negative exponent, squaring the components, and then simplifying by multiplying and adding the exponents.

Step-by-step explanation:

The expression given is (a^3b^{-4}/2a^{-1})^{-2} and we need to simplify it while keeping all exponents positive.

To simplify the expression, we'll first address the negative exponent on the outside by taking the reciprocal of the base and then raising everything inside to the power of two.

Next, we'll use the properties of exponents to simplify further.

When we take the reciprocal, the negative exponents inside will become positive, and vice versa.

Then raising to the power of 2 means we have to double every exponent inside the parentheses.

Here's the step-by-step simplification:

1. Starting with the original expression: (a^3b^{-4}/2a^{-1})^{-2}.
2. Take the reciprocal and square each component: (2a^{-1}/a^3b^{-4})^{2}.
3. Simplify inside the parentheses first: 2^2a^{-2}/a^6b^{-8}.
4. Multiply the exponents inside by 2 (since the whole term is squared): 2^{2} \times a^{-2}\times a^{6}\times b^{8}.
5. Simplify further by adding/subtracting the exponents: 4a^{(6-2)}b^{8} = 4a^{4}b^{8}.

So, the simplified expression is 4a^4b^8.