diff --git a/Grade 10/Math/MPM2DZ/Unit 5: Rational Expressions.md b/Grade 10/Math/MPM2DZ/Unit 5: Rational Expressions.md new file mode 100644 index 0000000..16ac50e --- /dev/null +++ b/Grade 10/Math/MPM2DZ/Unit 5: Rational Expressions.md @@ -0,0 +1,68 @@ +# Unit 5: Rational Expressions + +## Rational Exponents + +`Power Form:` $`\large a^{\frac{m}{n}}`$ + +`Radical Form:` $`n \sqrt{a^m} = (n \sqrt{a})^m`$ + +`Powers with negative rational exponents:` $`\large{a}^{\frac{-m}{n}} = \dfrac{1}{a^{\frac{m}{n}}} = \dfrac{1}{a^{n\sqrt{a^m}}}`$ + +`Powers with negative exponents:` $`a^-n = \dfrac{1}{a^n}`$ + +**Notes:** When dealing with power form, always reduce the exponent if you can. +- $`(-2)^{\frac{2}{4}} \rightarrow (-2)^{\frac{1}{2}}`$ + +## Solving Exponential Equations + +Eg. $`5^{x+1} = 125`$ + +1. Change expressions on both sides to the **SAME BASE** and simplify the exponent(s). $`5^{x+1} = 5^3`$ +2. Equate the exponents. $`{x+1} = 3`$ +3. Solve for the variable and checks solutions, if required. $`x+1 = 3 \implies x = 2`$ + +## Restrictions Of A Rational Expression + +Basically the denominator of a fraction can never be $`0`$. Therefore, we have to put restrictions on the variables of the denominator such that the final result of +the denominator is not equal to $`0`$. + +### Steps To Simplify Rational Expressions + +1. Factor fully +2. Divide common factor +3. State restrictions of **original** expression. + + +## Multiplying And Dividing Rational Expressions + +Multiplying: +1. Factor the numerators and denominators +2. State ALL restrictions on the variables +3. Using division, remove any common factors in the numerator and denominator +4. Multiply the numerators, then multiply the denominators +5. Write the result as a single expression + +Dividing: +1. Factor the numerators and denominators +2. State all restrictions on the variables +3. Take the reciprocal of the second rational expression and change the $`\divide`$ to $`\times`$ +4. State any NEW restrictions (When you try to flip the fraction, the denominators of the original and new fraction must be considered) +5. Using division, remove any common factors in the numerator and denominator +6. Multiply the numerators, then multiply the denominators +7. Write the result as a single expression + +## Adding And Subtracting Rational Expressions +1. Factor the denominator. +2. State the restrictions on the variables. +3. Determine the lowest common denominator. +4. Write each expression with the common denominator. +5. Add or subtract the numerators. (Combine them into one large expression) +6. Expand the numerator. +7. Simplify the numerator by combining like terms. +8. Factor the numerator, if factorable. +9. Divide out common factors. + +**Notes:** The LCD is not always the product of the denominators. To determine: +- Factor the denominators, find common coefficients, find a term that contains all the **UNIQUE** factors. + + diff --git a/Grade 10/Math/MPM2DZ/Unit 5: Rational.md b/Grade 10/Math/MPM2DZ/Unit 5: Rational.md deleted file mode 100644 index 67f99e8..0000000 --- a/Grade 10/Math/MPM2DZ/Unit 5: Rational.md +++ /dev/null @@ -1,5 +0,0 @@ -# Unit 5: Rational Expressions - -## Rational Exponents - -`Power Form:` $`\large a^{\dfrac{m}{n}}`$ \ No newline at end of file