# 1. 2x / 3-5 = ? 2. 42a4b2c / 28ab7c = ? 3. 3d2/4g - 5/2g2 = ?

## Expert Answers

Unfortunately, the way you have written the math text makes your questions unclear. Therefore, to answer your questions, I will rewrite them as follows. I hope that your intended meaning is retained.

1. \frac{2x}{3-5} = \frac{2x}{-2} = -x

The fraction \frac{2x}{3-5} is called an algebraic fraction. An algebraic fraction is a fraction that contains algebraic expressions in its numerator or denominator, or in both the numerator and the denominator. To simplify an algebraic fraction, identify like terms (common factors) in the numerator and the denominator and then cancel them out.

In the above fraction, however, we first simplify the denominator 3-5 to get -2 . The fraction then becomes \frac{2x}{-2}, where -2 is a common factor in both the numerator and denominator, thereby cancelling out to get -x , which is the answer.

2. \frac{42a^{4}b^{2}c}{28ab^{7}c}

To simplify this algebraic fraction, it is important to know the following laws of indices that apply to numbers that have similar bases:

-Product rule: x^{m} \cdot x^{n} = x^{m+n} (the two numbers being multiplied have the same base x )

-Quotient rule: x^{m} \div x^{n} = x^{m-n}

Thus, in the given algebraic fraction, we first find the common factors in the numerator and denominator. These are 14 , a , b^{2} , and c . 14 cancels out in the numerator and denominator to give \frac{3}{2} . a cancels out in the numerator and denominator to give a^{3} . Alternatively, using the quotient rule of indices, a^{4} \div a = a^{4-1} = a^{3} . b^{2} cancels out in the numerator and denominator to give \frac{1}{b^{5}} . Alternatively, using the Quotient rule of indices, b^{2} \div b^{7} = b^{2-7} = b^{-5} = \frac{1}{b^{5}} . Finally, c cancels out in the numerator and denominator to give 1. We multiply these results out to get \frac{3}{2}\cdot a^{3} \cdot \frac{1}{b^{5}}\cdot 1 = \frac{3a^{3}}{2b^{5}} .

3. \frac{3d^{2}}{4g} - \frac{5}{2g^{2}}

To simplify these algebraic fractions, we proceed as we would when subtracting two simple fractions. Thus, we first find the Least Common Multiple (LCM) of the denominators of the two fractions (4g and 2g^{2} ), which is 4g^{2} . Thus, the fraction simplifies to \frac{3d^{2}g - 10}{4g^{2}} .

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