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17 Cards in this Set
- Front
- Back
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What is the formula for the difference of two cubes?
(x³ - y³) = |
(x³ - y³) = (x - y)(x² + xy + y²)
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What is the formula for the sum of two cubes?
(x³ + y³) = |
(x³ + y³) = (x + y)(x² - xy + y²)
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Whenever you square both sides to solve a radical equation, you must check for:
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extraneous roots
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How do you check for extraneous roots?
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Plug the root back into the original equation. Extraneous roots will have no solution.
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What are the steps for solving a radical equation by squaring each side twice?
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1) Isolate the radicals on either side.
2) Square both sides and simplify. 3) Isolate and separate any remaining radicals and square again. 4) Check ALL solutions since extraneous roots can be produced by squaring. |
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x raised to the m/n power =
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ⁿ√xᵐ
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x⁻ᵐ/ⁿ =
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1/ⁿ√xᵐ
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What are the steps for solving an equation with radical exponents?
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1) Raise each side of the equation to the reciprocal power.
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What are the rules for solving an equation with radical exponents, ie.. xⁿ/ᵐ = k
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-There are two real even roots of any positive real number, one positive and one negative. ie... √16 = 4 or -4.
-There is exactly one real odd root of any real number. i.e... ³√8 = 2, but ³√-8 = -2. -If the exponent of x is even and k is > 0, then there will be two solutions: x = ±k -If the exponent of x is odd, there is only one real solution, x =k -If the exponent of x is even and k < 0, there is no real solution. |
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The odd roots of positive numbers are:
The odd roots of negative numbers are: |
positive
negative |
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How many real even roots do positive numbers have?
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2, one positive and one negative i.e... √4 = 2 and -2
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How many real odd roots does any number have?
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only 1. i.e... ³√8 = 2 but (-2)³ ≠ 8 it = -8
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When solving xᵐ/ⁿ = k:
-If the exponent is even and k is > 0, how many solutions will there be? |
then there will be two solutions: x = ±k
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When solving xᵐ/ⁿ = k:
-If the exponent is even and k < 0, how many solutions will there be? |
there is no real solution.
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When solving xᵐ/ⁿ = k:
-If the exponent is odd, how many solutions will there be? |
there is only one real solution, x =k
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What is an "equation of the quadratic type?"
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A higher degree equation that can be converted into a quadratic equation by substituting a single variable for the higher degree exponent. ie..
x⁴ -14x² + 45 = 0. Let u = x². So u² - u +45 = 0. |
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How do you solve an absolute value quadratic equation?
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1) Like always, place the value outside the absolute value on the other side of the equation
which will be = ± the value and drop the absolute value perimeter. 2) Factor the quadratic equations like normal. 3) Check each root for extraneous roots. |
