Mathematics Benchmarks, Grades K-12

Algebra II End-of-Course Exam Content Standards—Core: Operations on Numbers and Expressions (Priority: 15%)

Successful students will be able to perform operations with rational, real, and complex numbers, using both numeric and algebraic expressions, including expressions involving exponents and roots. There are a variety of types of test items including some that cut across the objectives in this standard and require students to make connections and, where appropriate, solve contextual problems.

O1. Real numbers

a. Convert between and among radical and exponential forms of numerical expressions.

  • Convert between expressions involving rational exponents and those involving roots and integral powers.

    Example: 5 to the three-halves = square root of 5-cubed = 5 times square root of 5; fourth root of 27 = fourth root of three-cubed = 3 to the three-fourth

b. Simplify and perform operations on numerical expressions containing radicals.

  • Convert radicals to alternate forms and use the understanding of this conversion to perform calculations with numbers in radical form.

    Example: square root of 8 + square root of 18 = 2 times square root of 2 + 3 times square root of 2 = 5 times square root of 2

c. Apply the laws of exponents to numerical expressions with rational and negative exponents to order and rewrite them in alternative forms.

  • Apply the properties of exponents in numerical expressions.

    Example: 35 · 32 = 3(5+2) = 37 = 2187, 3 to the fifths over 3-squared = 3(5-2) = 33 = 27, (35)2 = 3(5·2) = 310 = 59049, 3 to the 5-halves = open parentheses, square root of 3, close parentheses to the fifth = square root of 3 to the fifth = 9 times square root of 3

  • Convert between expressions involving negative exponents and those involving only positive ones.

    Example: 3-negative-squared = 1 over 3-squared = 1-ninth; 2 to the negative-third over 7 to the negative-first = 7 over 2 to the third = 7-eighths

O2. Complex numbers

a. Represent complex numbers in the form a + bi, where a and b are real; simplify powers of pure imaginary numbers.

  • Every real number, a, is a complex number because it can be expressed as a + 0i.
  • Represent the square root of a negative number in the form bi, where b is real; simplify powers of pure imaginary numbers.

    Example: square root of negative 8 = 2i times square root of 2

    Example: square root of 256 = 16i

    Example: i5 = -i

b. Perform operations on the set of complex numbers.

  • Add, subtract, and multiply complex numbers using the rules of arithmetic.
  • Divide complex numbers using conjugates.
  • This process can also be applied to division by irrational numbers involving square roots such as a + square root of b and a - square root of b.

    Example: fraction: 5 + 4i over 3 minus 2i; equals fraction: 5 + 4i over 3 minus 2i; times fraction: 3 + 2i over 3 + 2i; fraction: equals 15 + 22i + 8i squared, over 9 minus 4i squared; equals fraction: 7 + 22i over 13, or seven-thirteenths + 22-thirteenths times i

O3. Algebraic expressions

a. Convert between and among radical and exponential forms of algebraic expressions.

Example: x to the 3-halves = square root of x-cubed = x times square root of x

Example: square root of a to the sixth times b to the fourth = absolute value of a to the 6-halves times b to the 4-halves = absolute value of a-cubed times b-squared

b. Simplify and perform operations on radical algebraic expressions.

Example: square root of x-squared + 6x + 9 = square root of open parentheses, x + 3, close parentheses, squared = absolute value of x + 3

c. Apply the laws of exponents to algebraic expressions, including those involving rational and negative exponents, to order and rewrite them in alternative forms.

Example: a4 · a3 = a(4+3) = a7, a to the fourth over a-cubed = a(4-3) = a, (a4)3 = a(4·3) = a12

Example: y to the negative squared = 1 over y-squared, z to the 2-thirds = cube root of z-squared = open parentheses, cube root of z, close parentheses, squared

Example: r to the negative-three over t to the negative squared = t=squared over r-cubed, a to the negative fourth over b-cubed = 1 over a to the fourth times b-cubed, (a3b5)2 = a6b10

d. Perform operations on polynomial expressions.

  • Limit to at most multiplication of a binomial by a trinomial.
  • For division limit the divisor to a linear or factorable quadratic polynomial.
  • Division may be performed using factoring.

e. Perform operations on rational expressions, including complex fractions.

  • These expressions should be limited to linear and factorable quadratic denominators.
  • Complex fractions should be limited to simple fractions in numerators and denominators.

    Example: a + b over open parentheses, 1 over a + 1 over b, close parentheses, divided by ab = a + b over fraction b + a over ab, divided by ab = open parentheses, a + b, close parentheses, times ab over open parentheses, b + a, close parentheses, divided by ab = ab divided by ab = 1

f. Identify or write equivalent algebraic expressions in one or more variables to extract information.

Example: The expression, C + 0.07C, represents the cost of an item plus sales tax, while 1.07C is an equivalent expression that can be used to simplify calculations of the total cost.

Example: x minus fraction y-squared over x can be rewritten as fraction x-squared minus y-squared over x

About the Benchmarks

Elementary (K–6) Strands and Grade Levels

Secondary (7–12) Strands

Secondary Model Course Sequences

Secondary Assessments and Tasks

Correlations to the Secondary Benchmarks

Supporting Resources

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