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Which of these differences yields a rational expression? A. (3x^3+2x-7)/(sqrt (x)+3)-(4x^3-3)/(2sqrt (x)-7) B. (3sqrt [3](x)+2sqrt (x)-7)/(x+3)-(4sqrt [3](x)-3)/(2x-7) C. (3x^3+2x-7)/(x+3)-(4x^3-3)/(2x-7) D. (3x^frac (1)/(3)+2x^(1)/(2)-7)(x^(1)/(2)+3)-(4x^frac (1)/(3)-3)(2x^(1)/(2)-7)

Question

Which of these differences yields a rational expression? A. (3x^3+2x-7)/(sqrt (x)+3)-(4x^3-3)/(2sqrt (x)-7) B. (3sqrt [3](x)+2sqrt (x)-7)/(x+3)-(4sqrt [3](x)-3)/(2x-7) C. (3x^3+2x-7)/(x+3)-(4x^3-3)/(2x-7) D. (3x^frac (1)/(3)+2x^(1)/(2)-7)(x^(1)/(2)+3)-(4x^frac (1)/(3)-3)(2x^(1)/(2)-7)

Which of these differences yields a rational expression? A. (3x^3+2x-7)/(sqrt (x)+3)-(4x^3-3)/(2sqrt (x)-7) B. (3sqrt [3](x)+2sqrt (x)-7)/(x+3)-(4sqrt [3](x)-3)/(2x-7) C. (3x^3+2x-7)/(x+3)-(4x^3-3)/(2x-7) D. (3x^frac (1)/(3)+2x^(1)/(2)-7)(x^(1)/(2)+3)-(4x^frac (1)/(3)-3)(2x^(1)/(2)-7)

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MitchellElite · Tutor for 8 years

Answer

'C'

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## Step 1<br />A rational expression is defined as the ratio of two polynomials. Each option must be analyzed to determine if it fits this definition:<br /><br />## Step 2 <br />Option A: \((3x^3+2x-7)/(\sqrt{x}+3)-(4x^3-3)/(2\sqrt{x}-7)\)<br />### The denominators \(\sqrt{x}+3\) and \(2\sqrt{x}-7\) are not polynomials because they contain square roots of \(x\). Therefore, this expression is not a rational expression.<br /><br />## Step 3 <br />Option B: \((33\sqrt{x}+2\sqrt{x}-7)/(x+3)-(43\sqrt{x}-3)/(2x-7)\)<br />### Similar to Option A, this expression contains square roots in the numerators, making it not a rational expression.<br /><br />## Step 4 <br />Option C: \((3x^3+2x-7)/(x+3)-(4x^3-3)/(2x-7)\)<br />### Both the numerators and denominators are polynomials. This expression is a rational expression.<br /><br />## Step 5 <br />Option D: \((3x^{1/3}+2x^{1/2}-7)/(x^{1/2}+3)-(4x^{1/3}-3)/(2x^{1/2}-7)\)<br />### This expression contains fractional powers of \(x\) in both the numerators and denominators, which do not constitute polynomials. Therefore, this is not a rational expression.<br /><br />Thus, the only option that yields a rational expression is Option C.
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