Algebra and Geometry Review – Exercise 10

LESSON 1: Rational exponents (Unit fraction exponents and whole number bases)

Evaluate: 27^(1/3)
256^(1/4)

EXPLANATION: We can use the following to evaluate the exponential expressions
a^(1/n)=√(n&a)
We have 27^(1/3)=∛27. So, we need to find the cube root of 27
Checking some positive integers, we see that 3 is the cube root of 27.
1^3=1*1*1=1
2^3=2*2*2=8
3^3=3*3*3=27
So, 27^(1/3) = 3
We have 256^(1/4)=∜256. So, we need to find the fourth root of 256
Checking some positive integers, we see that 4 is the fourth root of 256
1^4=1*1*1*1=1
2^4=2*2*2*2=16
3^4=3*3*3*3=81
4^4=4*4*4*4=256
So, 256^(1/4) = 4

ANSWER: 27^(1/3) = 3, 256^(1/4) = 4

LESSON 2: Order of operations with integers and exponents

Evaluate ((-2)^3-4)^2-5*5

EXPLANATION: We must follow the rules for order of operations
Evaluate expressions within parentheses
Evaluate terms with exponents
Multiply and divide (from left to right)
Add and subtract (from left to right)
If any of the rules does not apply, we skip it
((-2)^3-4)^2-5*5=(-8-4)^2-5*5 Evaluate expression within parentheses - evaluate exponent term
=(-12)^2-5*5 Finish evaluating expression within parentheses
=144-5*5 Evaluate terms with exponents
=144-25 Multiply and divide from left to right
=119 Add and subtract from left to right

ANSWER: 119

LESSON 3: Rational exponents (Non-unit fraction exponent with a whole number base)

Simplify 16^(5/4)

EXPLANATION: We use the rules of exponents to simplify the expression
16^(5/4)= 16^((1/4*5) )
=16^(1/4)5  By the power of a power rule
=(∜16)^5 Because 16^(1/4)=∜16
=2^5 Because ∜16=2
=32

ANSWER: 32

LESSON 4: Greatest common factor of two multivariate monomials

Find the greatest common factor of these two expressions:
28w^5 y^3 v^4  and 12y^8 v^2

EXPLANATION: The GCF is the largest factor of the two expressions
We start by finding the GCF of the coefficients 28 and 12, which is 4
We then look at the variables that are common to both of the given expressions
The common variables are y and v
The lowest powers of these common variables are y^3 and v^2
The GCF is the product of these lowest powers and 4
GCF =4y^3 v^2

ANSWER: 4y^3 v^2

LESSON 5: Power rules with positive exponents (Multivariate quotients)

Simplify and write answer without parentheses: ((-7x^2)/z)^2

EXPLANATION: We simplify the expression as follows
((-7x^2)/z)^2=(-7x^2 )^2/z^2  Using the power of a quotient rule
=((-7)^2 (x^2 )^2)/z^2  Using the power of a product rule in the numerator
=(49(x^2 )^2)/z^2  Evaluating (-7)^2 to get 49
=(49x^4)/z^2 Using the power of a power rule in the numerator

ANSWER: (49x^4)/z^2 



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