Intermediate Algebra

What you'll learn

• Apply exponent rules (product, quotient, power, negative, and zero exponents)
• Interpret rational exponents as roots: x^(1/n) = the nth root of x, and x^(m/n) = the nth root of x^m
• Understand logarithms as the inverse of exponents: log_b(x) = y means b^y = x
• Identify and extend arithmetic and geometric sequences
• Find the nth term of arithmetic sequences: a_n = a_1 + (n−1)d
• Find the nth term of geometric sequences: a_n = a_1 × r^(n−1)
• Work with complex numbers (powers of i and a+bi arithmetic) and 2x2 matrices (add, scalar multiply, multiply, and take the determinant)

Key concepts

Exponent rules: x^a × x^b = x^(a+b), x^a / x^b = x^(a−b), (x^a)^b = x^(ab), x^0 = 1, x^(−n) = 1/x^n. Rational (fractional) exponents mean roots: x^(1/n) is the nth root of x, and x^(m/n) is the nth root of x^m (for example 8^(2/3) = the cube root of 8, squared, = 4). If you ever need to compare logs with different bases, the change-of-base rule log_b(x) = log(x) / log(b) lets you rewrite them. Logarithms reverse exponentiation: log₂(8) = 3 because 2³ = 8. Key log rules: log(ab) = log(a) + log(b), log(a/b) = log(a) − log(b), log(a^n) = n·log(a). An arithmetic sequence has a constant difference between terms (2, 5, 8, 11 with d = 3). A geometric sequence has a constant ratio (3, 6, 12, 24 with r = 2). Complex numbers: the imaginary unit i is defined by i² = −1 (so the square root of −1 is i). Its powers cycle in a period of four: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, and then the pattern repeats, so to simplify a high power of i, divide the exponent by 4 and use the remainder. A complex number is written a + bi, where a is the real part and b is the imaginary part. Add and subtract by combining like parts: (3 + 2i) + (1 − 5i) = 4 − 3i. Multiply using FOIL and then replace i² with −1: (2 + 3i)(1 − i) = 2 − 2i + 3i − 3i² = 2 + i + 3 = 5 + i. This i² = −1 rule is also why a negative discriminant produces complex (not real) solutions. Matrices: a matrix is a rectangular array of numbers with rows and columns; a 2x2 matrix has two of each. Add or subtract two matrices of the same dimensions entry by entry. Scalar multiplication multiplies every entry by the same number. To multiply two matrices, the number of columns of the first must equal the number of rows of the second, and each entry of the product is a row-times-column dot product; for two 2x2 matrices [[a, b], [c, d]] and [[e, f], [g, h]] the product is [[ae + bg, af + bh], [ce + dg, cf + dh]]. Matrix multiplication is not commutative, so order matters. The determinant of [[a, b], [c, d]] is ad − bc. These advanced topics typically appear in the later, harder questions of the Math section.

Pro tips

• If you see a logarithm question and freeze, convert it to exponential form: log_b(x) = y → b^y = x.
• For sequences, finding the common difference (arithmetic) or common ratio (geometric) is always the first step.
• Negative exponents mean 'reciprocal,' not 'negative number'. This is a common misconception the ACT exploits.