Pennsylvania Keystone Exam — Algebra I

Simplify the radical expression completely: \(\sqrt{72}\).

• A\(6\sqrt{2}\)
• B\(2\sqrt{6}\)
• C\(36\sqrt{2}\)
• D\(12\sqrt{2}\)

Simplify the monomial product: \((3x^4)(4x^5)\).

• A\(12x^{20}\)
• B\(7x^9\)
• C\(12x^9\)
• D\(7x^{20}\)

Simplify the expression using exponent rules: \((2x^3)^4\).

• A\(8x^{12}\)
• B\(16x^{12}\)
• C\(16x^7\)
• D\(8x^7\)

Express the number \(0.00045\) in standard scientific notation.

• A\(4.5 \times 10^{-4}\)
• B\(4.5 \times 10^{-5}\)
• C\(45 \times 10^{-5}\)
• D\(4.5 \times 10^4\)

Which of the following numbers is an irrational number?

• A\(\sqrt{16}\)
• B\(0.\overline{3}\)
• C\(\sqrt{20}\)
• D\(-\frac{7}{4}\)

Simplify the algebraic expression with negative exponents: \(\frac{24a^6b^{-2}}{4a^{-2}b^3}\).

• A\(\frac{6a^8}{b^5}\)
• B\(\frac{6a^4}{b^5}\)
• C\(6a^8b\)
• D\(\frac{6a^4}{b}\)

Simplify the expression by combining like radical terms: \(3\sqrt{20} - 2\sqrt{45} + \sqrt{80}\).

• A\(4\sqrt{5}\)
• B\(2\sqrt{5}\)
• C\(5\sqrt{5}\)
• D\(6\sqrt{5}\)

Estimate the value of \(\sqrt{130}\) to the nearest tenth.

• A\(11.1\)
• B\(11.4\)
• C\(11.7\)
• D\(12.1\)

Evaluate \((4 \times 10^5) \times (3 \times 10^{-8})\) and express the result in standard scientific notation.

• A\(1.2 \times 10^{-2}\)
• B\(1.2 \times 10^{-3}\)
• C\(12 \times 10^{-3}\)
• D\(1.2 \times 10^{-13}\)

Simplify the expression with rational exponents: \(\left(\frac{8x^9}{27y^6}\right)^{\frac{2}{3}}\).

• A\(\frac{4x^6}{9y^4}\)
• B\(\frac{2x^3}{3y^2}\)
• C\(\frac{4x^3}{9y^2}\)
• D\(\frac{16x^6}{81y^4}\)

Which list orders the numbers \(\{3\sqrt{2}, 2\pi, \sqrt{35}, 4.2\}\) from least to greatest?

• A\(4.2, 3\sqrt{2}, \sqrt{35}, 2\pi\)
• B\(3\sqrt{2}, 4.2, \sqrt{35}, 2\pi\)
• C\(4.2, 3\sqrt{2}, 2\pi, \sqrt{35}\)
• D\(3\sqrt{2}, 4.2, 2\pi, \sqrt{35}\)

Simplify the quotient of radicals: \(\frac{\sqrt{48x^5}}{\sqrt{3x}}\).

• A\(4x^2\)
• B\(16x^2\)
• C\(4x^4\)
• D\(2x^2\sqrt{3}\)

Simplify the complex expression completely: \(\frac{(2x^{-3}y^2)^3 \cdot (4x^4y^{-1})}{8x^{-2}y^3}\).

• A\(\frac{4y^2}{x^3}\)
• B\(\frac{4x^3}{y^2}\)
• C\(\frac{32y^2}{x^3}\)
• D\(\frac{4y}{x^3}\)

Rationalize the denominator of the fraction: \(\frac{6}{\sqrt{5} - \sqrt{2}}\).

• A\(2\sqrt{5} + 2\sqrt{2}\)
• B\(6\sqrt{5} + 6\sqrt{2}\)
• C\(\sqrt{5} + \sqrt{2}\)
• D\(2\sqrt{5} - 2\sqrt{2}\)

A research lab facility has a length of \(4.5 \times 10^4\) meters and a width of \(2.4 \times 10^3\) meters. What is the ratio of its length to its width expressed in standard decimal notation?

• A\(18.75\)
• B\(1.875\)
• C\(187.5\)
• D\(0.1875\)

Add the polynomials: \((3x^2 - 4x + 7) + (2x^2 + 5x - 3)\).

• A\(5x^2 + x + 4\)
• B\(5x^2 - 9x + 10\)
• C\(6x^2 + x + 4\)
• D\(5x^2 + 9x + 4\)

Subtract the polynomials: \((7x^2 - 3x + 2) - (4x^2 + 5x - 6)\).

• A\(3x^2 - 8x + 8\)
• B\(3x^2 + 2x - 4\)
• C\(3x^2 - 8x - 4\)
• D\(11x^2 + 2x - 4\)

What is the Greatest Common Factor (GCF) of the monomials \(12x^4y^2\) and \(18x^2y^3\)?

• A\(6x^2y^2\)
• B\(36x^4y^3\)
• C\(6x^4y^3\)
• D\(12x^2y^2\)

Factor the binomial completely: \(x^2 - 49\).

• A\((x - 7)(x + 7)\)
• B\((x - 7)^2\)
• C\((x + 7)^2\)
• D\((x - 49)(x + 1)\)

Multiply the two binomials: \((3x - 4)(2x + 5)\).

• A\(6x^2 + 7x - 20\)
• B\(6x^2 - 7x - 20\)
• C\(6x^2 + 23x - 20\)
• D\(6x^2 - 20\)

Factor the quadratic expression completely: \(x^2 - 9x + 20\).

• A\((x - 4)(x - 5)\)
• B\((x + 4)(x + 5)\)
• C\((x - 2)(x - 10)\)
• D\((x - 1)(x - 20)\)

Factor the trinomial completely: \(2x^2 + 7x + 3\).

• A\((2x + 1)(x + 3)\)
• B\((2x + 3)(x + 1)\)
• C\((x + 3)(x + 2)\)
• D\((2x + 6)(x + 1)\)

Expand and simplify the binomial square: \((2x - 3)^2\).

• A\(4x^2 - 12x + 9\)
• B\(4x^2 + 9\)
• C\(4x^2 - 9\)
• D\(4x^2 - 6x + 9\)

Factor the polynomial completely: \(3x^3 - 12x\).

• A\(3x(x - 2)(x + 2)\)
• B\(3x(x^2 - 4)\)
• C\(3(x^3 - 4x)\)
• D\((3x - 6)(x + 2)\)

What is the area of a rectangle with length \((3x + 2)\) units and width \((x - 5)\) units?

• A\(3x^2 - 13x - 10\)
• B\(3x^2 + 13x - 10\)
• C\(3x^2 - 10\)
• D\(4x - 3\)

Factor by grouping: \(x^3 + 3x^2 - 4x - 12\).

• A\((x + 3)(x - 2)(x + 2)\)
• B\((x^2 - 4)(x - 3)\)
• C\((x + 3)(x^2 + 4)\)
• D\((x - 3)(x - 2)(x + 2)\)

Find the simplified expression for the perimeter of a triangle with side lengths \((2x^2 - 3x + 1)\), \((x^2 + 4x - 5)\), and \((3x - 2)\).

• A\(3x^2 + 4x - 6\)
• B\(3x^2 + 7x - 6\)
• C\(6x^2 + 4x - 6\)
• D\(3x^2 - 4x + 6\)

Factor the quadratic expression completely: \(6x^2 - 11x - 10\).

• A\((2x - 5)(3x + 2)\)
• B\((2x + 5)(3x - 2)\)
• C\((6x - 5)(x + 2)\)
• D\((3x - 5)(2x + 2)\)

Solve the quadratic equation for \(x\) by factoring: \(4x^2 - 12x + 9 = 0\).

• A\(x = \frac{3}{2}\)
• B\(x = -\frac{3}{2}\)
• C\(x = 3\)
• D\(x = \frac{2}{3}\)

Factor the polynomial \(x^4 - 16\) completely over the set of real numbers.

• A\((x^2 + 4)(x - 2)(x + 2)\)
• B\((x^2 - 4)(x^2 + 4)\)
• C\((x - 2)^2(x + 2)^2\)
• D\((x - 2)^4\)

Solve the multi-step linear equation for \(x\): \(4x - 7 = 21\).

• A\(x = 7\)
• B\(x = 3.5\)
• C\(x = 14\)
• D\(x = 28\)

Solve for \(m\): \(3(m + 4) = 27\).

• A\(m = 5\)
• B\(m = 9\)
• C\(m = 11\)
• D\(m = 3\)

Solve the absolute value equation: \(|y - 3| = 7\).

• A\(y = 10\) or \(y = -4\)
• B\(y = 10\) only
• C\(y = 4\) or \(y = -10\)
• D\(y = -4\) only

Solve for \(x\): \(5(2x - 3) - 3x = 2(x + 4) + 2\).

• A\(x = 5\)
• B\(x = 4\)
• C\(x = 3\)
• D\(x = 6\)

Solve the geometric volume formula for height \(h\): \(V = \frac{1}{3}\pi r^2 h\).

• A\(h = \frac{3V}{\pi r^2}\)
• B\(h = \frac{V}{3\pi r^2}\)
• C\(h = \frac{3\pi r^2}{V}\)
• D\(h = 3V - \pi r^2\)

Solve for \(x\): \(2|3x - 1| + 4 = 16\).

• A\(x = \frac{7}{3}\) or \(x = -\frac{5}{3}\)
• B\(x = 2\) or \(x = -\frac{2}{3}\)
• C\(x = \frac{7}{3}\) only
• D\(x = 3\) or \(x = -2\)

A rideshare service charges a base fee of \(\$3.50\) plus \(\$2.25\) per mile traveled. If a rider's total bill was \(\$21.50\), how many miles did the trip cover?

• A\(8\) miles
• B\(7.5\) miles
• C\(9\) miles
• D\(8.5\) miles

Solve the fractional linear equation for \(x\): \(\frac{x + 3}{4} + \frac{2x - 1}{3} = 5\).

• A\(x = 5\)
• B\(x = 4\)
• C\(x = 6\)
• D\(x = 7\)

Solve the absolute value equation with expressions on both sides: \(|2x - 5| = |x + 4|\).

• A\(x = 9\) or \(x = \frac{1}{3}\)
• B\(x = 9\) only
• C\(x = -1\) or \(x = 9\)
• D\(x = \frac{1}{3}\) only

Two vehicles leave the same gas station at the same time, traveling in opposite directions along a straight highway. Vehicle A travels at \(55\) miles per hour and Vehicle B travels at \(65\) miles per hour. How many hours will it take for them to be \(360\) miles apart?

• A\(3\) hours
• B\(3.5\) hours
• C\(4\) hours
• D\(2.5\) hours

Solve the inequality: \(3x - 5 \le 10\).

• A\(x \le 5\)
• B\(x \ge 5\)
• C\(x \le 1.67\)
• D\(x \ge 1.67\)

Which ordered pair \((x, y)\) is the exact solution to the system of linear equations: \(x + y = 6\) and \(2x - y = 3\)?

• A\((3, 3)\)
• B\((4, 2)\)
• C\((5, 1)\)
• D\((2, 4)\)

Solve the compound inequality: \(-2 < x + 3 \le 7\).

• A\(-5 < x \le 4\)
• B\(-5 \le x < 4\)
• C\(1 < x \le 10\)
• D\(-2 < x \le 4\)

Solve the linear inequality for \(x\): \(-4(2x - 1) > 20\).

• A\(x < -2\)
• B\(x > -2\)
• C\(x < -3\)
• D\(x > 3\)