Goat Math

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 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

Algebra and Geometry Review – Exercise 08

LESSON 1: Degree and leading coefficient of a univariate polynomial

What are the leading coefficient and degree of the polynomial?
3-3x^2+6x

EXPLANATION: We first rewrite this polynomial in standard form.
We rearrange the terms so that the exponents on the variable decrease from left to right.
3-3x^2+6x=-3x^2+6x+3 standard form
The leading term is the first term when the polynomial is in standard form.
So, for this polynomial, the leading term is -3x^2

Leading coefficient
The coefficient of a term is the number multiplying the variable,
when there is no variable, the coefficient is the term itself
The leading coefficient is the coefficient of the leading term.
For our polynomial, the leading term is -3x^2. So, the leading coefficient is -3.

Degree
The degree of a term is the exponent of its variable,
when there is no variable, the degree is zero
The degree of a polynomial is the degree of its leading term
For our polynomial, the leading term is -3x^2. So, the degree of the polynomial is 2.

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 1: Factoring a quadratic with leading coefficient greater than 1 (Problem type 1)

Factor: 2x^2+17x+15

EXPLANATION: Method 1 - We will factor using a method often called Trial and Error.

The coefficient of x^2 is 2

Using whole numbers, only 1 and 2 have a product of 2

So we'll look for integers n and m that satisfy the following.

2x^2+17x+15 = (1x+m)(2x+n)

Using FOIL to expand the right-hand side of this equation, we get the following

nm=15 and n+2m=17

We'll find all integers m and n such that mn=15

Then we'll check to see if n+2m=17

QUESTION 1: Introduction to the GCF of two monomials

Find the greatest common factor of: 12x^2 and 8x^4

EXPLANATION:

The GCF of 12 and 8 is 4

The GCF of x^2 and x^4 is x^2

So, the GCF of 12x^2 and 8x^4 is 4x^2

ANSWER: 4x^2

QUESTION 2: Factoring a difference of squares in one variable

Factor: u^2 – 4

EXPLANATION: Here is the factoring formula for the difference of squares

A^2 – B^2 = (A + B)(A - B)

We can use this for the current problem

u^2 – 4 = u^2 – 2^2 Writing as A^2 – B^2, with A=u and B=2

= (u + 2)(u - 2) Using the difference of squares formula

ANSWER: (u + 2)(u - 2)

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

Labels