Algebra and Geometry Review – Exercise 09

LESSON 1: Adding rational expressions with common denominators and monomial numerators

Subtract and Simplify: 8/(b+3)-1/(b+3)

EXPLANATION: 
Note that 8/(b+3) and 1/(b+3) have the same denominator b+3
So, we subtract with the numerators and the denominator stays the same.
8/(b+3)-1/(b+3)=(8-1)/(b+3)=7/(b+3)

ANSWER: 7/(b+3)

LESSON 2: Introduction to simplifying a radical expression with an even exponent

Simplify and Assume that the variable represents a positive real number: √(y^10 )

EXPLANATION: By definition, √(y^10 ) is a positive number whose square equals y^10
Note that 〖(y^5)〗^2=y^5*y^5=y^(5+5)=y^10 
Therefore, the square of y^5 is y^10
And in this problem, we are assuming y is positive, so y^5 is also positive
So, we have the following: √(y^10 )=y^5
Or √(y^10 )=y^(10÷2)=y^5

ANSWER: y^5

LESSON 3: Using distribution with double negation and combining like terms to simplify (Multivariate)

Simplify -3x-5(-5y+2x)-4y

EXPLANATION: 
-3x-5(-5y+2x)-4y
=-3x+25y-10x-4y  Using distributive property to remove parentheses
=-3x+(-10x)+25y+(-4y) Using commutative property to rearrange terms
=-13x+21y  Combining like terms

ANSWER: -13x+21y  

LESSON 4: Rewriting an algebraic expression without a negative exponent

Rewrite the expression without using a negative exponent and Simplify as much as possible
1/(2p^(-2) )

EXPLANATION: For any nonzero number a and any whole number n, we have the following
Rule 1: a^(-n)=1/a^n Move a^(-n) to the denominator and make the exponent positive
Rule 2:  1/a^(-n) =a^n Move a^(-n) to the numerator and make the exponent positive
We need to rewrite 1/(2p^(-2) ) without a negative exponent
We use Rule 2 and move p^(-2) to the numerator, making the exponent positive
1/(2p^(-2) )=p^2/2

ANSWER: p^2/2

LESSON 5: Multiplying binomials with negative coefficients

Multiply and Simplify: (-8y+5)(5y-2)

EXPLANATION: We multiply as follows
(-8y+5)(5y-2)=-40y^2+16y+25y-10 Using FOIL
=-40y^2+41y-10 Simplifying

ANSWER: -40y^2+41y-10