Algebra and Geometry Review – Exercise 07

LESSON 1: Product rule with positive exponents (Multivariate)

Multiply and simplify your answer as much as possible:  5v^4 w^2*3w^8*2v

EXPLANATION: Grouping similar factors, we get the following
5v^4 w^2*3w^8*2v=(5*3*2) v^4 vw^2 w^8
Then, we simplify using the properties of exponents
30v^(4+1) w^(8+2)=30v^5 w^10

ANSWER: 30v^5 w^10

LESSON 2: Introduction to square root multiplication

Simplify √3*√7

EXPLANATION: We'll use the following property of square roots to simplify our expression

for any nonnegative numbers a and b: √a*√b=√ab 

We simplify as follows: 

√3*√7= √(3*7)  Using the product property of square roots

=√21 Multiplying under the square root sign

The number under the square root 21 has no perfect square factors other than 1

Therefore, √21  is in simplified radical form and is our answer.

ANSWER: √21

LESSON 3: Power rules with positive exponents (Multivariate products)

Simplify and write your answer without parentheses: (-7wu^3 )^2

EXPLANATION: We simplify the expression as follows

(-7wu^3 )^2=(-7)^2 w^2 (u^3 )^2 Using the power of a product rule

=49w^2 (u^3 )^2 Evaluating (-7)^2 to get 49

=49w^2 u^6 Using the power of a power rule

ANSWER: 49w^2 u^6

LESSON 4: Introduction to the product rule with negative exponents

Simplify and write your answer with a positive exponent only

v^(-5) 〖*v〗^(-2)

EXPLANATION: We'll be using the following rules for exponents

Product rule: For any number a and any integers m and n, we have the following

a^m*a^n=a^(m+n)

So, when multiplying powers with the same base, we add the exponents

Negative exponent rule: For any nonzero number a and any integer m, we have the following

a^(-m)=1/a^m 

Here is what we get for the problem

v^(-5) 〖*v〗^(-2)=v^(-5+(-2) ) By the product rule above

=v^(-7)=1/v^7  By the rule for a negative exponent

ANSWER: 1/v^7 

LESSON 5: Simplifying a ratio of multivariate monomials (Advanced)

Simplify (18v^2 w^4)/(27v^3 w^4 )

EXPLANATION: We simplify (18v^2 w^4)/(27v^3 w^4 ) as follows

(18v^2 w^4)/(27v^3 w^4 )=  (2v^2 w^4)/(3v^3 w^4 ) Canceling the common factor 9

=(2w^4)/(3vw^4 ) Canceling the common factor v^2

=2/3v Canceling the common factor w^4

ANSWER: 2/3v