2/6/2021 Sophia :: Welcome
Score 24/25
You passed this Milestone
24 questions were answered correctly.
1 question was answered incorrectly.
1
MILESTONE
Evaluate the following expression using the properties of logarithms.
RATIONALE
To evaluate this expression, recall that logarithmic expressions can be re-written as exponential expressions. We'll apply the
following relationship to each individual term in the expression.
If we have a logarithmic expression in the form log subscript b open parentheses y close parentheses equals x it can
be rewritten as b to the power of x equals y. Let's apply this to the first term, .
tells us that raised to some number, x, equals .
raised to the power of 1 is , so is equal to 1. We can repeat this process with the next term, .
tells us that raised to some number, x, equals .
raised to the power of is , so is equal to 2. Repeat this one more time for the last term, .
tells us that raised to some number, x, equals
raised to the power of is , so is Substitute the calculated in for the log expressions to evaluate.
Once the values are substituted, evaluate to addition.
The expression evaluates to 6.
CONCEPT
Introduction to Logarithms
2
Use the FOIL method to evaluate the expression:
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RATIONALE
FOIL stands for First, Outside, Inside, Last, and helps us remember how to multiply two binomials. Begin by multiplying the first two terms,
and 4.
times 4 equals . Next, multiply the two outside terms, and .
times equals -5 because when you multiply two terms with the same number under the radical symbol, the result is just that number.
Next, multiply the two inside terms, 2 and 4.
2 times 4 equals 8. Next, multiply the last two terms, 2and .
equals . Now we can combine all four parts.
After using FOIL, is equal to . Next, identify any like terms and combine.
We can combine the two constant terms, -5 and 8, to get 3. We can also combine the two radical terms, and , to get . The
expression is equal to 3 plus 2 square root of 5.
CONCEPT
Multiplying Radical Expressions
3
Divide the following expression.
RATIONALE
Start by rewriting the expression into multiple fractions with as the denominator. Remember to use the correct signs (addition or subtraction)
between the fractions.
Now that we have individual fractions, we can simplify each fraction. To do this, cancel out common factors in the numerator and denominator.
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Let's consider the first set, .
simplifies to because we can factor out from both terms. Next, consider the second set, .
simplifies to because we can factor out from both terms. Next, consider the third set, .
simplifies to 6 because we can factor out from both terms. The expression can be simplified to .
CONCEPT
Polynomials Divided by Monomials
4
The cost of a small business is given by the expression where is the number of units produced. The business will be profitable whenever its profit exceeds
its cost.
If the profit region is shaded in purple, which of the following graphs corresponds to the given situation?
https://waldenu.sophia.org/spcc/college-algebra-3/milestone_take_feedbacks/7797384 3/24
Score 24/25
You passed this Milestone
24 questions were answered correctly.
1 question was answered incorrectly.
1
MILESTONE
Evaluate the following expression using the properties of logarithms.
RATIONALE
To evaluate this expression, recall that logarithmic expressions can be re-written as exponential expressions. We'll apply the
following relationship to each individual term in the expression.
If we have a logarithmic expression in the form log subscript b open parentheses y close parentheses equals x it can
be rewritten as b to the power of x equals y. Let's apply this to the first term, .
tells us that raised to some number, x, equals .
raised to the power of 1 is , so is equal to 1. We can repeat this process with the next term, .
tells us that raised to some number, x, equals .
raised to the power of is , so is equal to 2. Repeat this one more time for the last term, .
tells us that raised to some number, x, equals
raised to the power of is , so is Substitute the calculated in for the log expressions to evaluate.
Once the values are substituted, evaluate to addition.
The expression evaluates to 6.
CONCEPT
Introduction to Logarithms
2
Use the FOIL method to evaluate the expression:
https://waldenu.sophia.org/spcc/college-algebra-3/milestone_take_feedbacks/7797384 1/24
,2/6/2021 Sophia :: Welcome
RATIONALE
FOIL stands for First, Outside, Inside, Last, and helps us remember how to multiply two binomials. Begin by multiplying the first two terms,
and 4.
times 4 equals . Next, multiply the two outside terms, and .
times equals -5 because when you multiply two terms with the same number under the radical symbol, the result is just that number.
Next, multiply the two inside terms, 2 and 4.
2 times 4 equals 8. Next, multiply the last two terms, 2and .
equals . Now we can combine all four parts.
After using FOIL, is equal to . Next, identify any like terms and combine.
We can combine the two constant terms, -5 and 8, to get 3. We can also combine the two radical terms, and , to get . The
expression is equal to 3 plus 2 square root of 5.
CONCEPT
Multiplying Radical Expressions
3
Divide the following expression.
RATIONALE
Start by rewriting the expression into multiple fractions with as the denominator. Remember to use the correct signs (addition or subtraction)
between the fractions.
Now that we have individual fractions, we can simplify each fraction. To do this, cancel out common factors in the numerator and denominator.
https://waldenu.sophia.org/spcc/college-algebra-3/milestone_take_feedbacks/7797384 2/24
, 2/6/2021 Sophia :: Welcome
Let's consider the first set, .
simplifies to because we can factor out from both terms. Next, consider the second set, .
simplifies to because we can factor out from both terms. Next, consider the third set, .
simplifies to 6 because we can factor out from both terms. The expression can be simplified to .
CONCEPT
Polynomials Divided by Monomials
4
The cost of a small business is given by the expression where is the number of units produced. The business will be profitable whenever its profit exceeds
its cost.
If the profit region is shaded in purple, which of the following graphs corresponds to the given situation?
https://waldenu.sophia.org/spcc/college-algebra-3/milestone_take_feedbacks/7797384 3/24
