Algebraic Fractions Unit 5 Flashcards

Algebraic Fractions Review

• Expressions raised to the power of zero, like (55x^2)^0, equal 1, regardless of the base.
• For the expression (2^-3)+(4^-3), the value simplifies to 9/64.
• The expression (4^0)+(6^-3) simplifies to 1 1/216, combining a constant and a power of a fraction.
• The expression (1)/[(2^0)-(2^1)] simplifies to -1, as 2^0 equals 1 and 2^1 equals 2.

Positive Exponents

• To express 4(x+y)^-1 with positive exponents, the formulation becomes 4/(x+y).
• Simplifying (x^6n)/(x^3n) yields x^3 since the exponents subtract (6-3).
• For [(5x^3)^-2]/(2), the result is 1/25x^6 after applying negative exponents.
• (8x^3)^-3 also simplifies to 1/512x^9 using negative exponent rules.

Reducing Algebraic Fractions

• The fraction (60x^3)/(12x) simplifies to 5x^2 with the exclusion x≠0.
• From (24x^2 yz^4)/(8x^3 y^8 z), the reduction yields -3z^3/xy^7 with exclusions x≠0, y≠0, z≠0.
• The expression (70x^2)/(100x) simplifies to 7x/10 with exclusion x≠0.
• (26x^2 y^6 z^8)/(4x^6 z^10) reduces to (13y^6)/(2x^4 z^2) with exclusions x≠0, z≠0.

Exclusion and Restrictions

• The fraction (x+7)/(x^2 +4m-21) simplifies to 1/(x-3), with restrictions x≠-7, x≠3.
• From (-x^2 +16x-63)/(x^2 -2x-35), the reduction results in (-x+9)/(x+5), with x≠7, x≠-5.
• (5x-2)/(5x-2) simplifies to 1 with exclusion x≠(2/5).
• The expression (x-x^4)/(x^2 +x+1) simplifies to x-x^2, with no exclusions.

Further Reductions

• The fraction [(x+y)^-6]/[(x+y)^-10] simplifies to (x+y)^4, with exclusions x≠-y, y≠-x.
• [(4x)/(5y)]*[(3a)/(4b)] results in 3ax/5by by multiplying numerators and denominators.
• For the product of two fractions, [(15x^2 y^3)/(8z)]*[(16z^5)/(35xy^4)], the result is (6xz^2)/(7y).

Factored Forms

• The expression [(x^2 -4x+4)/(x-6)]*[(x^2 -36)/(x^2 -8x+12)] simplifies to (x-2)(x-6)/(x-6).
• The product of fractions [(x+4)/(x^2 -1)]*[(x^2 +3x-28)/(x^2 -16)] results in (x+7)/(x+1)(x-1).
• Dividing [(x^2 y)/(xy^2)] by [(x^2)/(y)] yields 1/x.
• Multiplying and dividing {(y^2 -16)/[(y+4)(y+4)]}/{[7y-28]/[3y+12]} results in 3/7.
• For {(2x^2 -18)/(x^2 -8x+15)} / {(10x+30)/(5x-15)}, the resulting simplification is (x-3)/(x-5).