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Chapter 0: Problem 102

Factor and simplify each algebraic expression. $$-8(4 x+3)^{-2}+10(5 x+1)(4 x+3)^{-1}$$

Short Answer

Expert verified
The simplified and factored form of the expression is \[ \frac{50x+2}{4(2x+1.5)^2} \]

Step by step solution

01

Distribute the Negative Exponents

Rewrite the expression with negative exponents as fractions. The expression then becomes \[ -8 \div (4x+3)^2 + 10\cdot(5x+1) \div (4x+3) \]
02

Isolate the Common Factors

Factorize the expressions under the reciprocal to create a more navigable expression: \[ -8 \div [(2(2x+1.5))^2] + 10\cdot(5x+1) \div (2(2x+1.5)) \]
03

Further Simplify the Expression

Cancel out the twos in both denominators, and simplify the common factors in the denominators: \[ -8 \div (4(2x+1.5)^2) + 10\cdot(5x+1) \div (2(2x+1.5)) \]
04

Combine the Expressions

After simplifying the denominators, combine the two fractions into one so that they have a common denominator. The expression now looks like this: \[ \frac{-8+10(5x+1)}{(4(2x+1.5)^2)} \]
05

Simplify the Numerator

Simplify the numerator: \[ \frac{-8+50x+10}{4(2x+1.5)^2} \]
06

Final Simplification

Carry out the addition in the numerator to produce the final simplified expression: \[ \frac{50x+2}{4(2x+1.5)^2} \]

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$5+3(x-4)=8(x-4)=8 x-32$$

Factor Completely. $$x^{4}-5 x^{2} y^{2}+4 y^{4}$$

Explain how to factor \(3 x^{2}+10 x+8\)

Exercises \(142-144\) will help you prepare for the material covered in the next section. Use the distributive property to multiply: $$2 x^{4}\left(8 x^{4}+3 x\right)$$

Determine whether each statement is trueor false. If the statement is false, make the necessary change(s) toproduce a true statement. $$x^{2}+36=(x+6)^{2}$$
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