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# Basic Algebra

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)

### Word problem in Algebra about ages

3 years ago, Liz's age was 1/3 of her mother's age. Now, 7 years later, her age will be 1/2 of her mother's age. How old are Liz and her mother?

### Worm or Acme Thread Problem Set Up

Please see the attached file for full problem description. I would like to get some assistance in setting up this problem. A=1.645 and they give the answer of x as .56506.

### Logarithms and Exponents Rule

Logarithms and exponents. See attached file for full problem description. Q1: Express in log form 82=64 Q2: express in exponential form log4256=4 Q3: find unknown log327=x Q4: express as sum diff or multiple logarithm a. log314 b. log2 5/10 c. log5 (n3) Q5: express as log of single quantity a. long23 + log22

### Algebra - Composition

For the given functions f and g, find: (a) the composition of (f o g) (4)  (b) (g o f) (2)  (c)( f o f)(1)   (d) (g o g) (0) f(x) = x3/2; g(x) = 2 / x + 1  *note: x3/2 is actually a square root number but I couldn't figure out how to make the numbers small.

### Algebra - Composition of Functions

For the given functions f and g, find: (a)  the composition of (f o g) (4) (b) (g o f) (2)  (c)( f o f) (1)  (d) (g o g) (0) f(x) = 2x; g(x) = 3xsquared + 1

### Algebra - Functions and Domains

For the given functions f and g, find the following functions and state the domain of each. (a) f + g   (b)f - g  (c)f x g  (d) f / g f(x)= square root of x + 1;  g(x) = 2 / x

### Algebra - Operations of Functions and Domains

For the given functions f and g, find the following functions and state the domain of each. (a) f + g  (b)f - g  (c)f x g  (d) f / g f(x) = 2xsquared + 3; g(x) = 4xcubed + 1

### Algebra - Graphing a Function

Sketch the graph of each function.  Be sure to label three points on the graph. If f(x) = integral (2x), find: (a) f(1.2)  (b) f(1.6)  (c) f(-1.8)

### Cubic Equation - Remainder Theorem

Solve: x3 - 2x2+x=-5(x-1)2.

### Proof: Groups and Integers

Let m be the smallest positive integer such that @^m=E for all @eS_n. Show that m=lcm(2,3,4,5,...,n).

### Roots of a Quadratic Equation : Word Problem

Assume that the sum and product of two roots of a quadratic equation are 5 and 6, respectively. Find two roots.

### Verify Fubini's Theorem integral evaluated over equilateral triangle

I am asked to verify Fubini's Theorem for an integral evaluated over an equilateral triangle. I am asked to fully discuss the reasons for the limits of integration in my solution. See attached file for full problem description.

Factor: 9q^2-72q+750.

### Testing for Divisibility (Without using a Calculator!)

Test the following for divisibility by 2,3,4,5,6,8,9,and 10. (No calculator, explain your thinking). i. 6 543 210 b. What is the smallest whole number that is divisible by 2,3,4,5,6,8,9 and 10.

### Simplifying Expressions : Number Properties

Simplify each of the following expressions further and identify the number property(ies) used in each. 1. 2 x (15+8)= (2 x 15)+ (2 x 8)= 2. 5/2 x (3/4 x 2/5)= (5/2 x 2/5)x 3/4=

### Several proofs by induction

Show by induction that: a) Sum(n^2/[(2n-1)*(2n+1)],n=1..) = n(n+1)/[2*(2n+1)] b) Sum( r/(r-1)!, r=1..n) = 1-1/(n+1)! c) n^3+3n^2-10n is divisible by 3 d) 4^(2n+1) + 3^(n+2) is divisible by 13 See attached file for full problem description.

### Binomial Expansion : Finding Coefficients

Please see the attached file for the fully formatted problems. Find the following: a) The coefficient of x in the expansion of b) The coeffiecient of in the expansion of c) The coefficient of in the expansion of d) The coefficient of in the expansion of is equal to the coefficient of in the