Rules of ExponentsDate: 10/02/1999 at 18:24:22 From: CHRIS HELTON Subject: Rules of indices The question I am having trouble with is: Use the rules of indices to simplify the following expression: (16x^2)^(1/4)*(4x^3)^(1/2) -------------------------- (27^4)^(1/3) I don't know where to start breaking the question down to attempt to simplify it. This question is part of the foundation topic assignment given to me in the class titled Mathematics 1 at the University of Strathclyde. Date: 10/11/1999 at 21:29:59 From: Doctor Sandi Subject: Re: Rules of indices Hi Chris, First of all I want to go through some of the log laws with you. Consider 64 = 4^3 The 3 is called the exponent; the 4 is called the base. This can also be expressed as log(base 4) 64 = 3. Something else worth committing to your permanent memory is that any number raised to the power of 1/2 is the same as the square root of that number. For example, 4^(1/2) = sqrt(4) = 2. This also happens with numbers raised to the power of 1/3: the answer will be the same as the cube root of the number. Here are some rules for dealing with indices (write these out properly without the ^ so that they don't look so clumsy): 1. a^m * a^n = a^(m+n) 2. a^m / a^n = a^(m-n) 3. (a^m)^n = a^(mn) 4. (ab)^m = (a^m)(b^m) 5. a^0 = 1 6. a^(-n) = 1/(a^n), a is not equal to 0 7. a^(1/q) = q-root(a) 8. a^(p/q) = q-root (a^p) = (q-root(a))^p Generally when simplifying index expressions: 1. Express all terms in index form (lowest base) where necessary 2. Remove the brackets 3. Add and subtract indices 4. Express with positive indices Now to your question: [(16x^2)^(1/4)*(4x^3)^(1/2)]/[(27^4)^(1/3)]. When you have an index raised to the power of another index, you multiply the indices. You end up with: First the numerator (top half of the fraction) [(16^(1/4))(x^(2*(1/4))][(4^(1/2)(x^(3*1/2)] = [2x^(1/2)][2x^(3/2)] = (4x^[(1/2)+(3/2)] = (4x^2) then the denominator: (27^4)(^1/3) Let's deal with the power first. It is a power raised to another power so we will multiply them together and end up with: 4 * (1/3) = 4/3 So we have 27^(4/3) = 81 So the answer is (4x^2)/81 I hope that this has been helpful for you. If you have any other other questions that you would like to ask Dr. Math, please feel free to write back. - Doctor Sandi, The Math Forum http://mathforum.org/dr.math/ |
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