Question

Evaluate each expression. a. \(2 x-4(x-3),\) where \(x=\frac{1}{2}\) b. \(a(a-2)+a(3 a-2),\) where \(a=\frac{1}{3}\) c. \(2 x+6(x+2),\) where \(x=\frac{3}{4}\) d. \(6 x y-y^{2},\) where \(x=\frac{5}{16}\) and \(y=\frac{1}{8}\) e. \(4 x^{3}+x^{2},\) where \(x=\frac{1}{2}\)

Short answer

Question: Evaluate the following expressions: a. \(2 x-4(x-3),\) where \(x=\frac{1}{2}\) b. \(a(a-2)+a(3 a-2),\) where \(a=\frac{1}{3}\) c. \(2 x+6(x+2),\) where \(x=\frac{3}{4}\) d. \(6 x y-y^{2},\) where \(x=\frac{5}{16}\) and \(y=\frac{1}{8}\) e. \(4 x^{3}+x^{2},\) where \(x=\frac{1}{2}\) Answer: a. 11 b. -\(\frac{19}{9}\) c. 18 d. \(\frac{7}{32}\) e. \(\frac{3}{4}\)

Step-by-step solution

Substitute the given value of x

Replace x with \(\frac{1}{2}\) in the given expression: \(2(\frac{1}{2}) - 4(\frac{1}{2} - 3)\)

Simplify the expression

Calculate the values inside the parenthesis and multiplication: \(1 - 4(-\frac{5}{2}) = 1 + 10 = 11\) b. \(a(a-2)+a(3 a-2),\) where \(a=\frac{1}{3}\)

Substitute the given value of a

Replace a with \(\frac{1}{3}\) in the given expression: \(\frac{1}{3}(\frac{1}{3} - 2) + \frac{1}{3}(3\frac{1}{3} - 2)\)

Simplify the expression

Calculate the values inside the parenthesis and multiplication: \(\frac{1}{3}(\frac{-5}{3}) + (\frac{1}{3}) = -\frac{5}{9} - \frac{2}{3} = -\frac{19}{9}\) c. \(2 x+6(x+2),\) where \(x=\frac{3}{4}\)

Substitute the given value of x

Replace x with \(\frac{3}{4}\) in the given expression: \(2(\frac{3}{4}) + 6(\frac{3}{4}+2)\)

Simplify the expression

Calculate the values inside the parenthesis and multiplication: \(\frac{3}{2} + 6(\frac{11}{4}) = \frac{3}{2} + \frac{33}{2} = 18\) d. \(6 x y-y^{2},\) where \(x=\frac{5}{16}\) and \(y=\frac{1}{8}\)

Substitute the given values of x and y

Replace x with \(\frac{5}{16}\) and y with \(\frac{1}{8}\) in the given expression: \(6(\frac{5}{16})(\frac{1}{8}) - (\frac{1}{8})^{2}\)

Simplify the expression

Calculate the values inside the parenthesis and multiplication: \(\frac{15}{16} - \frac{1}{64} = \frac{14}{64} = \frac{7}{32}\) e. \(4 x^{3}+x^{2},\) where \(x=\frac{1}{2}\)

Substitute the given value of x

Replace x with \(\frac{1}{2}\) in the given expression: \(4(\frac{1}{2})^{3} + (\frac{1}{2})^{2}\)

Simplify the expression

Calculate the values inside the parenthesis and multiplication: \(4(\frac{1}{8}) + \frac{1}{4} = \frac{1}{2} + \frac{1}{4} = \frac{3}{4}\)