Question

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

Short answer

Question: Evaluate the following expressions for the given values of the variables: A. 2x - 4(x-3), where x = 1 1/2 B. a(a-2) + 3a(a+4), where a = 2/3 C. 2x + 6(x+2), where x = 2 1/4 D. 6xy - y^2, where x = 1 5/16 and y = 1/8 E. x^3 + x^2, where x = 1 1/2 Answer: A. 9 B. 76/9 C. 30 D. 125/64 E. 45/8

Step-by-step solution

Substitute the value of x

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

Convert mixed numbers to improper fractions

Convert \(1\frac{1}{2}\) to an improper fraction: \(1\frac{1}{2}=\frac{3}{2}\).

Simplify the expression

Using the order of operations (PEMDAS): \(2\left(\frac{3}{2}\right) - 4\left(\frac{3}{2} - 3\right)= 3 - 4\left(\frac{3}{2} - \frac{6}{2}\right)=3-4\left(-\frac{3}{2}\right)\).

Multiply and simplify

\(3-4\left(-\frac{3}{2}\right)=3+6=\boxed{9}\). B. Evaluate \(a(a-2)+3a(a+4),\) where \(a=\frac{2}{3}\).

Substitute the value of a

Replace a with \(\frac{2}{3}\) in the expression: \(\frac{2}{3}\left(\frac{2}{3}-2\right) +3\frac{2}{3}\left(\frac{2}{3}+4\right)\).

Simplify the expression

Using the order of operations (PEMDAS): \(\frac{2}{3}\left(\frac{2}{3} - \frac{6}{3}\right) + 2\left(\frac{2}{3} + \frac{12}{3}\right) = \frac{2}{3}\left(-\frac{4}{3}\right) + 2\left(\frac{14}{3}\right)\).

Multiply and simplify

\(-\frac{8}{9}+\frac{28}{3}=\frac{-8+84}{9}=\frac{76}{9}\). So, the expression evaluates to \(\boxed{\frac{76}{9}}\). C. Evaluate \(2x+6(x+2),\) where \(x=2 \frac{1}{4}\).

Substitute the value of x

Replace x with \(2\frac{1}{4}\) in the expression: \(2(2\frac{1}{4})+6(2\frac{1}{4}+2)\).

Convert mixed numbers to improper fractions

Convert \(2\frac{1}{4}\) to an improper fraction: \(2\frac{1}{4} = \frac{9}{4}\).

Simplify the expression

Using the order of operations (PEMDAS): \(2\left(\frac{9}{4}\right) + 6\left(\frac{9}{4} + \frac{8}{4}\right) = \frac{18}{4} + 6\left(\frac{17}{4}\right)\).

Multiply and simplify

\(\frac{18}{4}+\frac{102}{4}=\frac{120}{4}=\boxed{30}\). D. Evaluate \(6xy-y^2,\) where \(x=1\frac{5}{16}\) and \(y=\frac{1}{8}\).

Substitute the values of x and y

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

Convert mixed numbers to improper fractions

Convert \(1\frac{5}{16}\) to an improper fraction: \(1\frac{5}{16}=\frac{21}{16}\).

Simplify the expression

Using the order of operations (PEMDAS): \(6\left(\frac{21}{16}\right)\left(\frac{1}{8}\right) - \left(\frac{1}{64}\right) = \frac{126}{64}-\frac{1}{64}\).

Subtract and simplify

\(\frac{126}{64}-\frac{1}{64}=\frac{125}{64}\). So, the expression evaluates to \(\boxed{\frac{125}{64}}\). E. Evaluate \(x^3+x^2,\) where \(x=1\frac{1}{2}\).

Substitute the value of x

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

Convert mixed numbers to improper fractions

Convert \(1\frac{1}{2}\) to an improper fraction: \(1\frac{1}{2} = \frac{3}{2}\).

Simplify the expression

Using the order of operations (PEMDAS): \(\left(\frac{3}{2}\right)^3+\left(\frac{3}{2}\right)^2 = \frac{27}{8}+\frac{9}{4}\).

Add and simplify

\(\frac{27}{8}+\frac{18}{8}=\frac{45}{8}\). So, the expression evaluates to \(\boxed{\frac{45}{8}}\).