Title: 8. Algebraic fractions
Course: Demonstrating STACK (v4.3.7)

Simplify the following algebraic fractions: \[ \frac[value][value]. \]
Feedback: Remove the common factors from the top and bottom (in this case the a): \[ \frac[value][value] = [value]. \]

Simplify the following algebraic fractions: \[ \frac[value][value]. \]
Feedback: Remove the common factors from the top and bottom. \[ \frac[value][value] = [value]. \]

Simplify the following algebraic fractions: \[ \frac[value][value]. \]
Feedback: Remove the common factors from the top and bottom (in this case the [value]): \[ \frac[value][value] = [value]. \] The important thing is that you remember that \(\frac[value][value] = 1\) and so the top of the fraction does not cancel to 0.

Simplify the following algebraic fractions: \[ \frac[value]{-1}. \]
Leave you answer as a polynomial expression.
Feedback: The only fact required for this question is that dividing by \({-1}\) is the same as multiplying by \({-1}\). Therefore in this example: \[\frac[value]{-1} = [value]. \]

Simplify the following algebraic fraction: \[ \frac[value][value]. \]
Feedback: The first step should be to take a factor of \({-1}\) from the numerator of the fraction so that: \[ \frac[value][value] = \frac{-(a-b)}[value]. \] Now, it is clear that the factor \((a-b)\) can be cancelled from the numerator and the denominator of the fraction leaving [value].

Express the following as a single fraction in it's lowest terms: \[\frac[value][value]+\frac[value][value]. \]
Feedback: Adding algebraic fractions is exactly the same as adding purely numerical fractions. When the denominators are equal, as in this example, the addition is simple. All that is required is to add the numerators together and divide by the denominator. It may then require simplifying. In this example: \[\frac[value][value]+\frac[value][value] = [value]. \]

Express the following as a single fraction in it's lowest terms: \[\frac[value][value]+\frac[value][value]. \]
Feedback: Adding algebraic fractions is exactly the same as adding purely numerical fractions. You can use a general rule like: \[\frac{av}{c}+\frac{bv}{d} = \frac{adv+cbv}{cd} = \frac{(ad+cb)v}{cd}.\] Or you can convert the fractions so the denominator of both is the same (the LCM) and add the fractions as normal. In this example: \[\frac[value][value]+\frac[value][value] = \frac[value][value]+\frac[value][value] = [value]. \]

Simplify the following algebraic fraction: \[ \frac[value][value]. \]
Feedback: Remove the common factors from the top and bottom (in this case the \(([value])\): \[ \frac[value][value] = [value]. \] You may find it easier to substitute \(u = [value]\) and thereby remove the brackets and cancel the u's. However, make sure to leave your final answer in terms of [value].

Express the following as a single fraction in it's lowest terms: \[\frac[value][value]-\frac[value][value]. \]
Feedback: You can treat compound terms, \([value]\) as a single variable when simplifying the above expression. Alternatively you can substitute \(u = [value]\) and thereby simplify with a single variable: \[\frac[value][value]-\frac[value][value] = \frac[value][value]-\frac[value][value] = \frac[value][value] = [value]. \]

Express the following as a single fraction in it's lowest terms: \[\frac[value][value]+\frac[value][value]. \]
Feedback: You can treat compound terms, \([value]\) as a single variable when simplifying the above expression. Alternatively you can substitute \(u = [value]\) and thereby simplify with a single variable: \[\frac[value][value]+\frac[value][value] = \frac{[value]u}[value]+\frac{[value]u}[value] = \frac{[value]u}[value]+\frac{[value]u}[value] = \frac{[value]u}[value] = [value]. \]

Express the following as a single fraction in it's lowest terms: \[\frac[value][value]+1. \]
Feedback: When adding 1 to a fraction you need to express 1 as a fraction with a denominator equal to the other fraction in question. In this case \([value]\). So \( 1 = \frac[value][value]\). Now you have two fractions with equal denominators. Therefore you can add the fractions simply: \[\frac[value][value]+1 = \frac[value][value]+\frac[value][value] = [value]. \]

Express the following as a single fraction in it's lowest terms: \[\frac[value][value]-1. \]
Feedback: When subtracting 1 from a fraction you need to express 1 as a fraction with a denominator equal to the other fraction in question. In this case \([value]\). So \( 1 = \frac[value][value]\). Now you have two fractions with equal denominators. Therefore you can subtract 1 simply: \[\frac[value][value]-1 = \frac[value][value]-\frac[value][value] = [value]. \]

Express the following as a single fraction in it's lowest terms: \[\frac[value][value]+\frac[value][value]. \]
Feedback: When adding fractions with linear expressions the method remains the same. First, find a suitable common denominator and add the fractions. After adding, collect the 'like terms' and cancel any common factors from the numerator and the denominator. In this example: \[\frac[value][value]+\frac[value][value] = \frac[value][value]+\frac[value][value] = \frac[value][value] = [value]. \]

Express the following as a single fraction in it's lowest terms: \[\frac[value][value]+\frac[value][value]. \]
Feedback: When adding fractions with linear expressions the method remains the same. Firstly, find a suitable common denominator and add the fractions. After adding, collect the 'like terms' and cancel any common factors from the numerator and the denominator. In this example the linear expressions are on the denominator so the best option is to take [value] as the common denominator: \[\frac[value][value]+\frac[value][value] = \frac[value][value]+\frac[value][value] = \frac[value][value] = [value]. \]

Express the following as a single fraction in it's lowest terms: \[\frac[value][value]+\frac[value][value]. \]
Feedback: When adding fractions with linear expressions the method remains the same. Firstly, find a suitable common denominator and add the fractions. After adding, collect the 'like terms' and cancel any common factors from the numerator and the denominator. When you have linear expressions in the denominator it is usually best to take the common factor as the product of the denominators, \(\left(i.e. [value]\right)\). In this example: \[\frac[value][value]+\frac[value][value] = \frac[value][value]+\frac[value][value] = \frac[value][value] = [value]. \]

Express the following as a single fraction in it's lowest terms: \[\frac[value][value]+\frac[value][value]-\frac[value][value]. \]
Feedback: When dealing with more than one fraction you can either focus on just two fractions at a time or, alternatively, you can find a suitable common factor for all the fractions and then peform all the arithmetic in one step. Using the latter technique in this example: \[\frac[value][value]+\frac[value][value]-\frac[value][value] = \frac[value][value]+\frac[value][value]-\frac[value][value] = [value]. \]

Simplify the following: \[[value]\cdot[value].\]
Leave your answer as a single fraction it it's simplest form.
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Feedback: When multiplying algebraic fractions use the same method as for normal fractions. Multiply the numerators and the denominators together to get the numerator and denominator of the answer. Be sure to reduce the fraction to it's lowest terms. In this example: \[[value]\cdot[value] = \frac[value][value] = [value]. \]

Simplify the follwing: \[[value]\cdot[value].\]
Give your answer as a single fraction in it's lowest terms.
=
Feedback: When multiplying algebraic fractions use the same method as for normal fractions. Multiply the numerators and the denominators together to get the numerator and denominator of the answer. Be sure to reduce the fraction to it's lowest terms. In this example: \[[value]\cdot[value] = \frac[value][value] = [value]. \]

Simplify the following: \[[value]\cdot[value].\]
Leave your answer as a single fraction in it's simplest terms.
=
Feedback: When multiplying algebraic fractions use the same method as for normal fractions. Multiply the numerators and the denominators together to get the numerator and denominator of the answer. Be sure to reduce the fraction to it's lowest terms. In this example: \[[value]\cdot[value] = \frac[value][value] = [value]. \]

Simplify the following: \[[value]\div[value].\]
Leave your answer as a single fraction in its simplest form.
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Feedback: When dividing algebraic fractions the method is the same as with normal fractions. Turn one of the fractions upside down and multiply. Remember to reduce the fraction to it's lowest terms if required. In this example: \[ [value]\div[value] = [value]\cdot[value] = [value]. \]

Simplify the following: \[\left( \left([value]-\frac{1}[value] \right)^{2}+2\right)\div{\frac{1}[value]}.\]
Give an equivalent expression without brackets.
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Feedback: When required to peform combinations of multiplication or division the best trick is to focus on one operation at a time. In this example: \[ \left(\left([value]-\frac{1}[value]\right)^{2}+2\right)\div{\frac{1}[value]} = {\left([value]^{2}+\frac{1}{[value]^{2}}-2\right)}\div{\frac{1}[value]} = {\left([value]^{2}+\frac{1}{[value]^{2}}-2\right)}\cdot[value] = [value]. \]

Simplify the following: \[ \frac{\frac{b+1}{a}+1}{b}. \]
Leave your answer as a single fraction in it's simplest terms.
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Feedback: When required to peform combinations of multiplication or division the best trick is to focus on one operation at a time. In this example: \[ \frac{\frac{b+1}{a}+1}{b} = \frac{b+1}{ab}+\frac{1}{b} = \frac{b+1}{ab}+\frac{a}{ab} = \frac{a+b+1}{ab}. \]

Write the following as a rational expression: \[[value]\div[value].\]
Leave your answer as a single fraction in its simplest form.
=
Feedback: When dividing algebraic fractions the method is the same as with normal fractions. Turn the second of the fractions upside down and multiply. Remember to simplify the fraction to it's lowest terms, if required. In this example: \[ [value]\div[value] = [value]\cdot[value] = [value]. \]