Algebra and Geometry Review – Exercise 05

QUESTION 1: Multiplying a univariate polynomial by a monomial with a positive coefficient

Use the distributive property to remove the parentheses and Simplify your answer as much as possible:

                10b^2(4b + 2b^5)

EXPLANATION: We use the distributive property and then simplify, as follows

                10b^2(4b + 2b^5) = 10b^2 * 4b + 10b^2 *2b^5   Distributing 10b^2 across the parentheses

                = (10*4)b^2*b + (10*2)b^2*b^5                Grouping similar factors

                = (10*4)b^3 + (10*2)b^7               Using the product rule for exponents

                = 40b^3 + 20b^7

ANSWER: 40b^3 + 20b^7

QUESTION 2: Square roots of perfect squares with signs

Evaluate the following and write "Not a real number" if applicable.

                Minus root 25: -sqrt(25)

                Root minus 36: sqrt(-36)

EXPLANATION: In these problems, we must deal with negative signs and square roots.

                We have that sqrt(a) is a real number only if a is positive or 0

                If a is negative, then sqrt(a) is not a real number

                We will use this fact in the current problem

                Note: sqrt(25) = 5, then mean that -sqrt(25) = -5

               

                Sqrt(-36) : If a is negative, then sqrt(a) is not a real number. So, Sqrt(-36) is not a real number

ANSWER:

                -sqrt(25) = -5

                sqrt(-36) = Not a real number

QUESTION 3: Simplifying a ratio of univariate monomials

                Simplify: 56w^4 / 7w^7

EXPLANATION: We simplify 56w^4 / 7w^7 as follows:

                56w^4 / 7w^7 = 8w^4 / w^7       Canceling the common factor 7

                = 8 / w^3             Canceling the common factor w^4

ANSWER: 8 / w^3

QUESTION 4: Evaluating an expression with a negative exponent

Rewrite the following without an exponent: (-8)^-2

EXPLANATION: For any non-zero number a and any whole number n, we have the following.

                a^-n = 1/a^n

                We'll apply this rule to (-8)^-2

                (-8)^-2 = 1/(-8)^2

                = 1 / (-8)*(-8)

                = 1/64

ANSWER: 1/64

QUESTION 5: Simplifying a sum or difference of two univariate polynomials

                Simplify: (v^2 – 3v + 7) + (-4v^2 + 6v +2)

EXPLANATION: (v^2 – 3v + 7) + (-4v^2 + 6v +2), we must add with expressions in parentheses

To add with expressions in parentheses, we can simply remove the parentheses.

Then, we combine like terms if possible.

(v^2 – 3v + 7) + (-4v^2 + 6v +2) = v^2 – 3v + 7 + -4v^2 + 6v +2

= v^2 – 4v^2 – 3v + 6v + 7 + 2      Focusing on like terms

= -3v^2 + 3v + 9                 Combining like terms

ANSWER: -3v^2 + 3v + 9

QUESTION 6: Simplifying the square root of a whole number less than 100

                Simplify: sqrt(20)

EXPLANATION: One of the properties of square roots is the following

                Sqrt(a*b) = sqrt(a)*sqrt(b)

                We'll use this property to simplify sqrt(20)           

First, we look for the greatest factor of 20 that is a perfect square.

20 = 4*5, where 4 is a perfect square

Then, we use the property to get the following:

Sqrt(20) = sqrt(4)*sqrt(5)

                = 2*sqrt(5)

We have that 2*sqrt(5) is in simplified radical form.

Then, the number under the square root 5, has no perfect square factors (other than 1).

ANSWER: 2*sqrt(5)

QUESTION 7: Square root of a rational perfect square

                Simplify and write your answer in simplest form: sqrt(49/25)

EXPLANATION: One of the properties of square roots is the quotient property

                Sqrt(a/b) = sqrt(a)/sqrt(b)            for any positive real numbers a and b

                Using this property, we can do the current problem

                sqrt(49/25) = sqrt(49)/sqrt(25)   By the quotient property

                = 7/5     because sqrt(49)=7, and sqrt(25)=5

ANSWER: 7/5

Algebra and Geometry Review – Exercise 05