## 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:

#### 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:

#### 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: 3^{5} · 3^{2} = 3^{(5+2)} = 3^{7} = 2187, = 3^{(5-2)} = 3^{3} = 27, (3^{5})^{2} = 3^{(5·2)} = 3^{10} = 59049,

- Convert between expressions involving negative exponents and
those involving only positive ones.
Example:

### 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* + 0*i*.
- Represent the square root of a negative number in the form
*bi*, where *b* is real; simplify powers of pure imaginary numbers.Example:

Example:

Example: *i*^{5} = -*i*

#### b. Perform operations on the set of complex numbers.

### O3. Algebraic expressions

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

Example:

Example:

#### b. Simplify and perform operations on radical algebraic expressions.

Example:

#### 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: *a*^{4} · a^{3} = a^{(4+3)} = a^{7}, = a^{(4-3)} = a, (a^{4})^{3} = a^{(4·3)} = a^{12}

Example:

Example: , (*a*^{3}b^{5})^{2} = a^{6}b^{10}

#### 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:

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

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

Example: can be rewritten as