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?
Perhaps I can help you with first few steps, and you continue~ For the first question, you need to apply the identity \(a^2-b^2=(a+b)(a-b)\) \( \ \frac{x^4-81}{x+3}=\frac{(x^2-9)(x^2+9)}{x+3}=\frac{(x-3)(x+3)(x^2+9)}{x+3}=...?\)
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, \[\frac{(x^2yz)^2(xy^2z^2)}{(xyz)^2}\]\[=\frac{x^{2\times 2 +1}y^{1\times 2+2}z^{1\times 2 +2}}{(xyz)^2}\]\[=\frac{x^{5}y^{4}z^{4}}{(xyz)^2}\]\[=\frac{x^{5}y^{4}z^{4}}{(xyz)^2}\]\[=x^{5-2}y^{4-2}z^{4-2}\]\[=...?\]
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.
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 :)
@callisto is 1. x^3 + 3x^2 - 9x - 27?
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