Operations with Complex Fractions Flashcards (20 cards)

What is the formula for multiplying algebraic fractions?

$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$

What is the first step in multiplying algebraic fractions?

Multiply the numerators and denominators.

What do you do after multiplying numerators and denominators in algebraic fractions?

Factor the resulting expressions.

What is the final step when multiplying algebraic fractions?

Cancel common factors.

What is the formula for dividing algebraic fractions?

$\frac{a}{b}\div\frac{c}{d}=\frac{ad}{bc}$

What is the first step in dividing algebraic fractions?

Multiply by the reciprocal of the divisor.

What should be done after multiplying by the reciprocal when dividing algebraic fractions?

Factor the expressions.

What is the final step in dividing algebraic fractions after factoring?

Simplify common factors.

What is the first step to simplify a complex fraction?

Perform the subtraction in its numerator and denominator.

What do you do after subtracting numerator and denominator in a complex fraction?

Divide the resulting fractions and simplify.

What is the simplified form of $\frac{\frac{1}{x}-\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}}$?

$\frac{xy}{x+y}$

What do complex fractions analyze in automotive mechanics?

Relationships among forces, movements, resonance frequencies, suspension, steering, electrical circuits, and engines.

What formula relates suspension resonance frequency to mass and spring stiffness?

$f=\frac{1}{2\pi}\sqrt{\frac{k}{m}}$

How is $x^2-4$ factored in the example $\frac{x-2}{2x} \cdot \frac{3x}{x^2-4}$?

As $(x-2)(x+2)$.

What is the simplified result of $\frac{x-2}{2x} \cdot \frac{3x}{x^2-4}$ after factoring and canceling?

$\frac{3}{2(x+2)} = \frac{3}{2x+4}$.

What is the simplified result of $\frac{x^2+2x+1}{3x^3}\div\frac{2x+2}{x^2}$?

$\frac{x+1}{6x}$

What algebraic identity is used to simplify complex fractions involving squares?

The difference of squares identity.

What is the formula for the difference of squares used in simplification?

$y^2 - x^2 = (y - x)(y + x)$

What do fractions model in vehicle suspension systems?

The relationship between forces and movements considering damper resistance, spring stiffness, and oscillation frequency.

How can suspension resonance frequency be adjusted?

By modifying the spring constant or the vehicle mass.