Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

If
x=3+2sqrt2, then find whether
x+(1)/(x) is rational or irrational.

If $x=3+2 \sqrt{2}$ , then find whether $x+\frac{1}{x}$ is rational or irrational.

View step-by-step help

View step-by-step help

Transformations of linear functions

Full solution

Q. If

x=3+2 \sqrt{2}

, then find whether

x+\frac{1}{x}

is rational or irrational.

Find $1/x$ Value: First, let's find the value of $1/x$ . $\newline$ $1/x = 1/(3 + 2\sqrt{2})$
Rationalize Denominator: To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator. $\newline$ $(1 \times (3 - 2\sqrt{2}))/((3 + 2\sqrt{2}) \times (3 - 2\sqrt{2}))$
Calculate Denominator: Now, calculate the denominator using the difference of squares formula. $\newline$ $(3 + 2\sqrt{2}) \times (3 - 2\sqrt{2}) = 3^2 - (2\sqrt{2})^2$
Perform Squaring: Perform the squaring of the terms. $3^2 - (2\sqrt{2})^2 = 9 - 4 \times 2$
Simplify Expression: Simplify the expression. $9 - 4 \times 2 = 9 - 8 = 1$
Write Simplified Expression: Now, write down the simplified expression for $1/x$ . $\newline$ $1/x = (3 - 2\sqrt{2})/1$
Add $x$ and $\frac{1}{x}$ : Add $x$ and $\frac{1}{x}$ . $\newline$ $x + \frac{1}{x} = (3 + 2\sqrt{2}) + (3 - 2\sqrt{2})$
Combine Like Terms: Combine like terms. $\newline$ $(3 + 2\sqrt{2}) + (3 - 2\sqrt{2}) = 3 + 3$
Final Simplification: Simplify the final expression. $\newline$ $3 + 3 = 6$

More problems from Transformations of linear functions

Question

This equation shows how the time required to screen-print a batch of shirts is related to the number of shirts in the batch.

\newline

t = s + 13

\newline

s

represents the number of shirts in the batch, and the variable

t

represents the time required to screen-print the shirts. How long does it take to screen-print a batch of

4

\newline

_________minutes

Posted 3 months ago

Question

Krysta wants to take piano lessons. She asked a teacher how many new pieces she could expect to learn each month. The teacher gave an estimate of

3

. Write an equation that shows how the total number of pieces Krysta would learn,

y

, depends on the number of months of piano lessons,

x

\newline

y

Posted 3 months ago

Question

Tom bought a boat for

\$3,000

and incurs a daily storage fee of

\$15

. Write a linear function to represent the total amount Tom will owe,

p(x)

x

days to pick it up.

\newline

Posted 4 months ago

Question

180

-page book cover to cover in a single session, at a constant rate. It took him

6

\newline

\newline

y

represent the number of pages left to read after

\newline

t

\newline

Which of the following information about the graph of the relationship is given?

\newline

1

\newline

\newline

x

\newline

\newline

y

\newline

(C) Slope and a point that is not an intercept

\newline

\newline

x

\newline

y

\newline

\newline

y

-intercept and a point that is not an intercept

\newline

(F) Two points that are not intercepts

Posted 4 months ago

Question

A charity organization had to sell

18

tickets to their fundraiser just to cover necessary production costs. They sold each ticket for

\$45

y

represent the net profit (in dollars) when they have sold

x

tickets. Which of the following information about the graph of the relationship is given?

\newline

1

\newline

x

\newline

y

\newline

(C) Slope and a point that is not an intercept

\newline

x

y

\newline

y

-intercept and a point that is not an intercept

\newline

(F) Two points that are not intercepts

Posted 2 months ago

Question

Hiro painted his room. After

3

hours of painting at a rate of

8

square meters per hour, he had

28

square meters left to paint.

\newline

\newline

y

represent the area (in square meters) left to paint after

\newline

x

\newline

Which of the following information about the graph of the relationship is given?

\newline

1

\newline

\newline

x

\newline

\newline

y

\newline

(C) Slope and a point that is not an intercept

\newline

\newline

x

\newline

y

\newline

\newline

y

-intercept and a point that is not an intercept

\newline

(F) Two points that are not intercepts

Posted 2 months ago

Question

Naoya read a book cover to cover in a single session, at a rate of

55

pages per hour. After

4

350

pages left to read.

\newline

y

represent the number of pages left to read after

x

\newline

Complete the equation for the relationship between the number of pages left and number of hours.

\newline

y=

Posted 1 month ago

Question

A teacher purchased

300

colored pencils for an upcoming project. The pencils she ordered come in packs of

20

30

14

packs in all. Write a system of equations to find the number of

20

(x)

30

(y)

the art teacher ordered.

Posted 2 months ago

Question

C=0.25(L-1200)

\newline

C

, in dollars, to the holder of a liability insurance account given a total liability of

L

dollars is given by the equation as long as

C

is positive. By how many dollars does the liability increase when the cost to the holder increases by

1

Posted 2 months ago

Question

P=\frac{2}{5} m+1.5

\newline

P

, in dollars, of a train ticket used to travel a distance of

m

miles is given by the equation. By how many miles does the traveling distance increase if the ticket price increases by

1

Posted 4 months ago