answer:To solve this problem, we sum the exponents of 10: begin{align*} (10^{0.6})(10^{0.4})(10^{0.3})(10^{0.2})(10^{0.5}) &= 10^{0.6 + 0.4 + 0.3 + 0.2 + 0.5} &= 10^{2.0} &= boxed{100}. end{align*}

answer:Let's denote the number of single burgers as S and the number of double burgers as D. We know that: 1. The total cost of the burgers is 68.50. 2. Single burgers cost 1.00 each, and double burgers cost 1.50 each. 3. The total number of burgers is 50. We can set up two equations based on this information: Equation 1 (total cost): 1.00S + 1.50D = 68.50 Equation 2 (total number of burgers): S + D = 50 We can solve these equations simultaneously to find the values of S and D. From Equation 2, we can express S in terms of D: S = 50 - D Now we can substitute S in Equation 1 with the expression we found from Equation 2: 1.00(50 - D) + 1.50D = 68.50 Expanding the equation, we get: 50 - D + 1.50D = 68.50 Combining like terms, we get: 50 + 0.50D = 68.50 Now, let's solve for D: 0.50D = 68.50 - 50 0.50D = 18.50 Divide both sides by 0.50 to find D: D = 18.50 / 0.50 D = 37 So, Caleb bought boxed{37} double burgers.

answer:The equation you've provided is a polynomial equation with two variables, x and y. It is not a linear equation, and there is no unique solution for x and y. This is because there are infinitely many pairs of (x, y) that can satisfy the equation. To find specific solutions, we would need additional information or constraints on x or y. For example, if we had another equation involving x and y, we could potentially solve the system of equations for specific values of x and y. Without additional information, we cannot find unique values for x and y that satisfy the equation x^3 + y^2x - 6x + 5y + 1 = boxed{0} .

answer:We are given the following information: 1. The age difference between two siblings is 3 years. 2. One year ago, the father's age was double the sum of the siblings' current ages. 3. In twenty years, the father's age will be the sum of the ages of the two siblings. We need to determine the current ages of the siblings. Let's denote: - The current age of the older sibling as ( x ). - The current age of the younger sibling as ( y ). - The current age of the father as ( z ). From the problem statement, the following equations can be formulated: 1. ( x - y = 3 ) (because the age difference between the siblings is 3 years) 2. ( z - 1 = 2(x + y) ) (one year ago, the father's age was double the sum of the siblings' current ages) 3. ( z + 20 = x + 20 + y + 20 ) (in twenty years, the father's age will be the sum of the siblings' ages) Let's simplify these equations step by step. First, from equation (3): [ z + 20 = x + y + 40 ] Subtract 20 from both sides: [ z = x + y + 20 ] Now we have the simplified system of equations: [ (1) quad x - y = 3 ] [ (2) quad z - 1 = 2(x + y) ] [ (3) quad z = x + y + 20 ] We solve this system step by step: 1. Substitute ( z ) from equation (3) into equation (2): [ (x + y + 20) - 1 = 2(x + y) ] Simplify: [ x + y + 19 = 2(x + y) ] Subtract ( x + y ) from both sides: [ 19 = x + y ] 2. Now, we have: [ (1) quad x - y = 3 ] [ (4) quad x + y = 19 ] Next, we solve for ( x ) and ( y ): Add equations (1) and (4): [ (x - y) + (x + y) = 3 + 19 ] [ 2x = 22 ] [ x = 11 ] Substitute ( x ) back into equation (4): [ 11 + y = 19 ] [ y = 8 ] Thus, the current ages of the siblings are: [ x = 11 quad text{(older sibling)} ] [ y = 8 quad text{(younger sibling)} ] To find the father's age ( z ), substitute ( x ) and ( y ) back into equation (3): [ z = x + y + 20 ] [ z = 11 + 8 + 20 ] [ z = 39 ] Finally, we conclude that: - The age of the older sibling is ( 11 ) years. - The age of the younger sibling is ( 8 ) years. - The age of the father is ( 39 ) years. [boxed{x = 11, , y = 8, , z = 39}]