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# Basic Algebra

### Cryptography: RSA Problem - Public and Private Keys

Fill in the blank choose public or private. Alice chooses two large prime numbers p and q. She finds their product m = pq, which is _____. She also finds n= (p-1)(q-1) which is _____. She chooses e which is _____and finds d=_____. The number e is public or private. and the number d is public or private. When Bo

### Plane Separation : Half Plane and Non-Collinear Points

If H1 is a half plane determined by the line l, prove that H1 has at least 3 noncollinear points.

### Betweenness and Elementary Figures: Three Collinear Points

Let A = (x1,y1), B = (x2,y2), and C = (x,y) be three collinear points in the Euclidean Plane with x1<x2. Prove that A-C-B iff x1<x<x2.

### Log Function : Integral and Derivative

Let g(x)=integral from 1 to x sin t ln t dt. Find g'(x)

### Sides of a Cube Problem

Eight small cubes are put together to form one large cube. All six sides of the larger cube are painted, the paint is allowed to dry, and then the cube is taken apart. a) How many of the small cubes will have paint on just one side? On two sides, On three sides? On no sides? b) Complete the following table, assuming in turn

### Least Common Multilpes

A. Find the LCM (Least common multiple) (84, 108). Show how you obtained your answer. B. Suppose LCM (18,A) = 72. What are the possible values, if any, for A? Explain your answer.

### Equivalence and Simplification

A. Show that 693/858 and 42/52 are equivalent in THREE different ways. B. Simplify 3^5 X 24^3 divide by 12^3 X (6^3)^2 ^ means exponent.

### Perform the Indicated Operations: Factoring

3x/x^2-4 + 4/x^3+8 Please try to factor this equation.

### Algebra : Solve the Rational Equation

(3/4)x + 5/6 = (4/3)x

### Quadratic Equation: Solve for X

2x OVER x+4 = 3 OVER x-1.

### Algebra : Graphing Functions

Please see the attached file for the fully formatted problems. Can you please assist me with the problems listed below? P. 237 1. a) #6, b) #18 2. a) #22, b) #24 3. a) #32, b) #46 P.253 4. a) #1, b) #2, c) #3. d_ #4 5. a) #10, b) #12 6. a) # 28, b) #30 7. P.260, Matched Problem 1 P.271 8. # 2 - 22 (Eve

### Domain and Range Function Found

Let f(x)=sqrt(2*x-6) . Find the largest possible domain for f . Find its range. Also graph the function.

### Algebraic simplification problem after a u-substitution derivation

The attached file contains a function that needs to have the first order derivative taken. I took the derivative, but I am making a mistake in the algebra simplification of the result. Would you please take a look. I am working on a dynamic programming problem, and one part of it requires that I take the derivative of the

### Linear Congruences: Example Problems

Find the solutions of (1) 153x is congruent to 6 (mod 12) (2) x + 1 is congruent to 3 (mod 7) (3) 8x is congruent to 6 (mod 422) (4) 363x is congruent to 345 (mod 624).

### Linear Congruences ( Part I)Incongruent Solutions

Find the solutions of (a) 4x is congruent to 3(mod 7) (b) 9x is congruent to 11(mod 26) (c) 3x + 1 is congruent to 4(mod 5) (d) 8x is congruent to 6(mod 14).

### Intervals Real Numbers

Please see the attached file for full problem description Find ten real numbers in the closed interval [0,1] such that for each value m from 2 to 10, the numbers lie so that there is one of them in each of the open intervals [Note that these conditions for m = 1,2......,10 all have to hold simultaneously for your

### Proving that an equality is false.

I have two sets of 64 numbers (1.1 to 7.4). Both number sets are created using the same equation for values of i from 0 to 63. m = 1.1 + ( i * 0.1 ) n = 1.1 + ( i * 0.1 ) I am trying to understand if the following equality is false in all cases except when the terms in each expression are equal (e.g. m^-12 = n^-12 and

The graph of the quadratic equation x^T Ax = [0,0,1]x, where A = [ 1/(alpha^2) 0 0 ] [ 0 -1/(beta^2) 0 ] [ 0 0 0 ] is a(n): A. ellipse B. hyperbola C. elliptic paraboloid D. parabolic cone E. hyperbolic paraboloid

### Inner Products for Unit Vectors

Let f1, f2, f3 be unit vectors in R3 such that < f1, f2 >= 1/2. Give a necessary and sufficient relationship between x =< f1, f3 > and y =< f2, f3 >. Please be sure to be rigorous and as detailed as possible.

E^x*e^x+1=e^x-1

3^(2x-1)=5x

10^x=200

5^x-8=23

### Solve logx - log2 = 5 for x.

Solve for x in the following logarithmic equation. Please show a detailed step by step solution. logx - log2 = 5

### Algebra Simplification: Expression with Multiple Exponents

Simplify the Expression [ (3(x^4)(y^2))/(2(x^7)(y^6))^3 x ((2x^-2)(y^4))^5 / ((3x^3)(y^-2))^5 ]^1/2 Make sure to show all steps.

### Algebra with Indices

Given that S=(R+D)^(1/3) and T=(R-D)^(1/3) where D= square root(R^2+Q^3), show that ST=-Q

### Square Root Functions Explained

Hi, can someone please explain to me how sqrt (square root of)128 becomes 8sqrt2 ; or sqrt68 becomes 2sqrt17? ; or sqrt45 becomes 3sqrt5? Thank you

### Simplifying Expression Function

Simplify the following expressions: a. 3x-2/2x2+x-3 Divide by 9x-6/2x-2 b.(108)1/2- (48)1/2+ (192)1/2 (should be exponent 1/2) c. (20x3+3x2-4x+5)/(4x2+3x-7) (should be exponent 3 and 2,2).

### Factoring Functions Solved

Factor the following: a. 81a4- 16b4 (should be exponent 4) b. 12x2+ x- 35 (should be exponent 2)