Algebra Flashcards (Aleks)

Multiplicative Property of Equality

• If ( 9 = -\frac{3}{4}y ), then multiplying both sides by the reciprocal (-\frac{4}{3}) leads to solving ( -12 = y ).

Simplifying Ratios of Monomials

• For ( \frac{30y^4}{5y^7} ), divide coefficients and subtract exponents: ( \frac{30}{5} = 6 ) and ( y^{4-7} = \frac{1}{y^3} ), resulting in ( \frac{6}{y^3} ).

Solving Linear Equations with Variables on Both Sides

• The equation ( -6(3x-1) + 4x = 1 + 2(2x-5) ) requires the distributive property, combining like terms, and isolating ( x ) to find ( x = \frac{5}{6} ).

Evaluating Expressions with Exponents

• Any number raised to the power of zero equals one, ( a^0 = 1 ).

Simplifying Ratios of Factored Polynomials

• For ( \frac{56(u-7)(2u-3)}{7(2u-3)(u+8)} ), cancel the common factor ( (2u-3) ) to simplify to ( \frac{8(u-7)}{u+8} ).

Basic Square Root Multiplication

• The product property states ( \sqrt{a} \sqrt{b} = \sqrt{ab} ), simplifying gives results like ( 4\sqrt{3} ) from ( \sqrt{48} ).

Signed Fraction Division

• For ( \frac{3}{20} \div \left(-\frac{21}{16}\right) ), take the reciprocal, cancel common factors leading to ( -\frac{4}{35} ).

Solving Equations in Factored Form

• In ( (w + 7)(w - 8) = 0 ), set each factor to zero: solutions are ( w = -7 ) and ( w = 8 ).

Factoring Quadratics with Leading Coefficient Greater Than 1

• For ( 14x^2 + 9x - 5 ), find integers ( p ) and ( q ) such that ( pq = -70 ) (from ( 14 \times -5 )) and ( p + q = 9 ).

Evaluating Functions

• Use substitution property in functions involving linear, quadratic, or cubic equations to analyze outputs.

Advanced Square Root Multiplication

• To simplify ( 2\sqrt{72} \sqrt{98} ), first simplify each square root, leading to answer ( 168 ) through ( \sqrt{7056} ).

Simplifying Multivariate Expressions

• With ( 3x - 4(4y - 3x) + 4y ), apply distributive property, combine like terms, and simplify to yield ( 15x - 12y ).

Rationalizing Denominators

• For ( \frac{\sqrt{2}}{\sqrt{3}} ), multiply numerator and denominator by ( \sqrt{3} ) to simplify the result to ( \frac{\sqrt{6}}{3} ).

Solving Two-Step Equations with Signed Fractions

• With ( \frac{3}{2}w - \frac{1}{7} = -\frac{8}{5} ), eliminate fractions using LCD, apply the distributive property, and isolate ( w ) leading to ( w = -\frac{34}{35} ).

Simplifying Ratios of Polynomials Using GCF

• Reducing ( \frac{8w^2 - 12w}{4w^2 - 16} ) involves factoring both numerator and denominator to arrive at ( \frac{2w - 3}{w - 4} ).

Degree and Leading Coefficient of a Polynomial

• The polynomial (-5 - 10w + w^6) has a degree of ( 6 ) (highest exponent) and a leading coefficient of ( 1 ).

Simplifying Products with Square Roots

• In the expression ( \sqrt{7}(\sqrt{6} - 11) ), apply distributive property and the product property of square roots for simplification.

Finding the LCM of Monomials

• For monomials ( 12a^4 ) and ( 9a^2 ), the LCM is determined as ( 36a^4 ) combining the LCM of coefficients and highest powers of variables.