For Problems 1 to 4, write the algebraic expression.
a. Twelve increased by three times a number.
b. The difference between a number and five.
a. Eight decreased by twice a number.
b. Six less than the total of a number and ten.
a. The product of three more than a number and the number.
b. Sum of ten times a number and fifteen.
a. Sum of fifteen and half of a number.
b. Product of two times a number and seven.
For Problems 5 to 20, simplify the algebraic terms, and express the answer with a positive exponent.
a. [latex] (-x)^2 \cdot (x)^{-4} [/latex]
b. [latex] (-x)^3 \cdot (x)^4 [/latex]
a. [latex] (-x)^3 \cdot (x)^{-6} [/latex]
b. [latex] x^3 \cdot (-x)^4 [/latex]
a. [latex] \displaystyle{\frac{x^8}{x^4}} [/latex]
b. [latex] \displaystyle{\frac{x^{-6}}{x^{-4}}} [/latex]
a. [latex] \displaystyle{\frac{x^7}{x^5}} [/latex]
b. [latex] \displaystyle{\frac{x^{-2}}{x^{-3}}} [/latex]
a. [latex] \displaystyle{\left(\frac{x^3}{y^{-2}}\right)^{\frac{1}{2}}} [/latex]
b. [latex] \sqrt[3]{x^6} [/latex]
a. [latex] \displaystyle{\left(\frac{x^{-3}}{y^{-2}}\right)^{\frac{2}{3}}} [/latex]
b. [latex] \sqrt[4]{x^{10}} [/latex]
a. [latex] \displaystyle{\frac{x^9}{x^5 \cdot x^2}} [/latex]
b. [latex] \displaystyle{\frac{x^{-4}}{x^5 \cdot x^{-9}}} [/latex]
a. [latex] \displaystyle{\frac{x^5 \cdot x^2}{x^3}} [/latex]
b. [latex] \displaystyle{\frac{x^{-5}}{x^4 \cdot x^{-3}}} [/latex]
a. [latex] (x^4)(3x^3) [/latex]
b. [latex] \displaystyle{\frac{x^6}{x^2}} [/latex]
a. [latex] (x^3)(2x^5) [/latex]
b. [latex] \displaystyle{\frac{x^9}{x^6}} [/latex]
a. [latex] \displaystyle{\left(\frac{x2}{y}\right)\left(\frac{x}{2y^2}\right)^2} [/latex]
b. [latex] \displaystyle{\frac{2x^{-4}y^6}{x^3y^{-4}}} [/latex]
a. [latex] \displaystyle{\left(\frac{2x^2}{y}\right)\left(\frac{3x}{y^2}\right)^2} [/latex]
b. [latex] \displaystyle{\frac{12x^{-5}y^2}{6x6y{-8}}} [/latex]
a. [latex] \displaystyle{\left(\frac{4x^3}{2y^2}\right)^3} [/latex]
b. [latex] \displaystyle{(8x^6)^{\frac{1}{3}}} [/latex]
a. [latex] \displaystyle{\left(\frac{3x^2}{4y^3}\right)^2} [/latex]
b. [latex] \displaystyle{\left(\frac{x^0}{y^3}\right)^3} [/latex]
a. [latex] \displaystyle{\left(\frac{x^7}{x^0}\right)^2} [/latex]
b. [latex] \displaystyle{\left(\frac{3x^{-2}y^7}{6x^3y^{-5}}\right)^2} [/latex]
a. [latex] \displaystyle{(4x^2)^{\frac{1}{2}}} [/latex]
b. [latex] \displaystyle{\left(\frac{3x^{-3}y^4}{6xy^{-2}}\right)^{-3}} [/latex]
For Problems 21 to 26, simplify the expressions, then evaluate for the given value of the variables in the brackets.
a. [latex] -4x^2 + 3x - 5 + 7x^2 - 2x + 3 [/latex] [latex] (x = 2) [/latex]
b. [latex] 4x^2 - 5 + 7x - 2x^2 - x - 3 [/latex] [latex] (x = -1) [/latex]
a. [latex] 3x^2 - x + 2 + x^2 - 5x - 2 [/latex] [latex] (x = 3) [/latex]
b. [latex] -5y^2 - 7y + 3 + y^2 - 5y + 2 [/latex] [latex] (y = -2) [/latex]
a. [latex] -y^2 + 4xy + x^2 - 6y^2 - xy - 11x^2 [/latex] [latex] (x = 1, y = 2) [/latex]
b. [latex] (x - 4)(x + 2) + 3(x + 2) [/latex] [latex] (x = 3) [/latex]
a. [latex] -4x^2 + 6xy - 6y^2 + 6x^2 - 2xy + 3y^2 [/latex] [latex] (x = 2, y = 1) [/latex]
b. [latex] (y - 2)(y - 3) + 2(y - 2) [/latex] [latex] (y = 4) [/latex]
a. [latex] (2x - 3)^2 - (x + 3)^2 [/latex] [latex] (x = 4) [/latex]
b. [latex] (5 + x)^2 + (4 - x)(4 + x) [/latex] [latex] (x = 5) [/latex]
a. [latex] (2x + 1)^2 - (x - 2)^2 [/latex] [latex] (x = 1) [/latex]
b. [latex] (3 - x)^2 + (x - 3)(x + 3) [/latex] [latex] (x = 2) [/latex]
For Problems 27 and 28, factor the expressions using the GCF, then evaluate for the given value of the variables in the brackets.
a. [latex] 6x^2 - 4x [/latex] [latex] (x = 1) [/latex]
b. [latex] 7xy + 14x^2 [/latex] [latex] (x = 3, y = 2) [/latex]
a. [latex] 8y^2 - 64y [/latex] [latex] (y = 2) [/latex]
b. [latex] 15y^2 + 10xy [/latex] [latex] (x = -1, y = 2) [/latex]
For Problems 29 and 30, factor the expressions by grouping.
a. [latex] 6x^3 - 2x^2 + 15x - 5 [/latex]
b. [latex] xy - 3y + 5x - 15 [/latex]
a. [latex] 6x^3 + 12x^2 + 3x + 6 [/latex]
b. [latex] y^2 - xy + 2y - 2x [/latex]
For Problems 31 and 32, factor the differences of squares.
a. [latex] 4x^2 - 9 [/latex]
b. [latex] 25x^2 - 64y^2 [/latex]
a. [latex] 1 - 16x^2 [/latex]
b. [latex] 81x^2 - 144y^2 [/latex]
For Problems 33 to 36, factor the trinomials.
a. [latex] x^2 + 5x - 36 [/latex]
b. [latex] 4x^2 + 16x + 15 [/latex]
a. [latex] x^2 - 4x - 77 [/latex]
b. [latex] 5x^2 + 20x - 60 [/latex]
a. [latex] x^2 + 16x + 64 [/latex]
b. [latex] 9x^2 - 24x + 16 [/latex]
a. [latex] 4x^2 + 4x + 1 [/latex]
b. [latex] x^2 - 8xy + 7y^2 [/latex]
For Problems 37 to 40, write the algebraic equation and solve.
a. Seventeen more than five times a number is forty-two.
b. A number divided by fifteen is forty-five.
a. The product of five and a number is seventy-five.
b. Three more than two times a number is nine.
a. The difference between a number and ten is ten.
b. The product of four times a number and three is thirty-six.
a. The sum of two times a number and eight is one hundred.
b. A number divided by three is seven.
For Problems 41 to 44, solve for the unknown variable, x, using the principles of equality.
a. [latex] 5x - 5 = 10 [/latex]
b. [latex] \displaystyle{\frac{x}{3} + 4 = 10} [/latex]
a. [latex] 3x - 5 = -17 [/latex]
b. [latex] \displaystyle{\frac{x}{4} - 2 = 1} [/latex]
a. [latex] 12 - 3x = 3 - 4x [/latex]
b. [latex] 4(x + 4) = 24 [/latex]
a. [latex] 4x - 2 = 13 - 6x [/latex]
b. [latex] 3(2x - 5) = 3 [/latex]