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Mathematics 21 Online
OpenStudy (anonymous):

1. What is the simplified form of x^4-81/x+3? 2. What is the simplified form of (x^2yz)^2(xy^2z^2)/(xyz)^2? 3.What is the excluded value of the rational expression 2x+6/4x-8? 4. What is the simplified form of x^2-25/x^2-3x-10?

OpenStudy (callisto):

Perhaps I can help you with first few steps, and you continue~ For the first question, you need to apply the identity \(a^2-b^2=(a+b)(a-b)\) \( \ \frac{x^4-81}{x+3}=\frac{(x^2-9)(x^2+9)}{x+3}=\frac{(x-3)(x+3)(x^2+9)}{x+3}=...?\)

OpenStudy (callisto):

For the second question, you need the following properties of exponential function: \[1. \ (a^m)^n = a^{mn}\]\[2. \ a^m \ times a^n = a^{m+n}\]\[3. \ \frac{a^m}{a^n} = a^{m-n}\]So, for your question, \[\frac{(x^2yz)^2(xy^2z^2)}{(xyz)^2}\]\[=\frac{x^{2\times 2 +1}y^{1\times 2+2}z^{1\times 2 +2}}{(xyz)^2}\]\[=\frac{x^{5}y^{4}z^{4}}{(xyz)^2}\]\[=\frac{x^{5}y^{4}z^{4}}{(xyz)^2}\]\[=x^{5-2}y^{4-2}z^{4-2}\]\[=...?\]

OpenStudy (callisto):

3. excluded value of the expression means the value that makes the denominator =0 So, put denominator of the fraction = 0 4x-8 = 0 <- solve x.

OpenStudy (callisto):

For the last question, to simplify the expression, you need to do factorization. Here's the identity, which is same as the first question, that is useful in the question\(a^2 - b^2 = (a-b)(a+b)\) To factorise the expression x^2 -25, you can apply the above identity. x^2 - 25 = x^2 - (5)^2 = (x-5)(x+5) To factorise the expression x^2-3x-10, you can do it in this way: x^2+2x - 5x -10 = x(x+2) -5 (x+2) = (x-5)(x+2) So, you can get this: \[\frac{ x^2-25}{x^2-3x-10} = \frac{ (x-5)(x+5)}{(x-5)(x+2)}\]Now, cross out the common factor and you'll get the answer :)

OpenStudy (anonymous):

@callisto is 1. x^3 + 3x^2 - 9x - 27?

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