Simplify the expression using the properties of exponents. Expand any numerical portion of your answer and only include positive exponents.
(3a^2)(4a^−1)^2(19b^2)(3b^−1)
To simplify the expression using the properties of exponents, we can follow these steps:
Step 1: Simplify each term separately.
First, simplify (4a^−1)^2:
(4a^−1)^2 = 4^2 * (a^−1)^2 = 16 * a^−2
Next, simplify 3b^−1:
3b^−1 = 3 * (b^−1) = 3 * (1/b) = 3/b
Step 2: Combine the simplified terms.
Now we can combine all the terms:
(3a^2)(4a^−1)^2(19b^2)(3b^−1) = (3a^2)(16 * a^−2)(19b^2)(3/b)
Step 3: Multiply and simplify.
Multiply the coefficients and combine the variables:
(3 * 16 * 19 * 3)(a^2 * a^−2)(b^2 * 1/b) = 216 * a^(2−2) * b^(2−1)
Simplifying exponents:
216 * a^0 * b^1 = 216 * 1 * b = 216b
Therefore, the simplified expression is 216b.