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How do you simplify #[6x^(-4/3) * 2x^(2/3)] /( 2x^(-1/3))# leaving only positive exponents?

Answer 1

Julia Adams

#6/x^(1/3)#

#[6x^(-4/3) * 2x^(2/3)] /( 2x^(-1/3))#

For ease of working, separate the coefficients from the variables - only the variables have indices to be attended to!

=#[6 xx color(blue)(x^(-4/3)) xx cancel2xx x^(2/3)] /( cancel2 xx color(red)(x^(-1/3))) " "x^-m = 1/x^m#

#[6 xx color(red)(x^(1/3)) xx x^(2/3)] /(color(blue)( x^(4/3)))#

=#[6 xx x^(3/3)] /( x^(4/3)) " "1/3+2/3 = 3/3#

=#6/x^(1/3)" "4/3 - 3/3 = 1/3#

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Answer 2

Julia Adams

To simplify the expression [6x^(-4/3) * 2x^(2/3)] / (2x^(-1/3)) leaving only positive exponents, you first need to apply the rules of exponents. By combining like terms and simplifying, the expression becomes:

[6x^{\left(-\frac{4}{3}\right)} \times 2x^{\left(\frac{2}{3}\right)} \div 2x^{\left(-\frac{1}{3}\right)} = 3x^{\left(-\frac{4}{3}+\frac{2}{3}-\left(-\frac{1}{3}\right)\right)} = 3x^{\left(-\frac{4}{3}+\frac{2}{3}+\frac{1}{3}\right)} = 3x^{\left(-\frac{4}{3}+\frac{3}{3}\right)} = 3x^{\left(-\frac{1}{3}\right)}]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.