All categories

2 years ago

Show that 30.1111... is a rational number.

1 answer:

8 0

We know that 30.1111... is a rational number because it ends in repeating decimals. We can show this by writing the number as a fraction using algebra.

x = 30.1111....

10x = 301.1111....

Subtract both sides by x

10x - x = 301.1111.... - 30.1111...

9x = 271

x = 271/9

The number can be rewritten as 271/9, which you can verify is the same number by checking on a calculator.

You might be interested in

Find the derivative of <img src="https://tex.z-dn.net/?f=tan%5E%7B-1%7D%20x" id="TexFormula1" title="tan^{-1} x" alt="tan^{-1} x

sladkih [1.3K]

$\huge{\color{magenta}{\fbox{\textsf{\textbf{Answer}}}}}$

$\frak {\huge{ \frac{1}{1 + {x}^{2} } }}$

Step-by-step explanation:

$\sf let \: f(x) = { \tan }^{ - 1} x \\ \\ \sf f(x + h) = { \tan}^{ - 1} (x + h)$

$\sf f'(x) = \frac{f(x+h) - f(x) }{h}$

$\sf \implies \lim_{ h \to 0 } \frac{ { \tan }^{ - 1}(x + h) - { \tan}^{ - 1}x }{h} \\ \\ \\ \sf \implies \lim_ {h \to 0} \frac{ { \tan}^{ - 1} \frac{x + h - x}{1 + (x + h)x} }{h}$

By using

$\sf { \tan}^{ - 1} x - { \tan}^{ - 1} y = \\ \sf { \tan}^{ - 1} \frac{x - y}{1 + xy} formula$

$\sf \implies \large \lim_{h \to0 } \frac{ { \tan}^{ - 1} \frac{h}{1 + hx + {x}^{2} } }{h} \\ \\ \\ \sf \implies \large{\lim_{h \to0} } \frac{ { \tan}^{ - 1} \frac{h}{1 + hx + {x}^{2} } }{ \frac{h}{1 + hx + {x}^{2} } \times (1 + hx + {x}^{2} )} \\ \\ \\ \sf \implies \large \lim_{h \to0} \frac{ { \tan}^{ - 1} \frac{h}{1 + hx + {x}^{2} } }{ \frac{h}{1 + hx + {x}^{2} } } + \lim_{h \to0} \frac{1}{1 + hx + {x}^{2} }$

Now putting the value of h = 0

$\sf \large \implies 0 + \frac{1}{1 + 0 + {x}^{2} } \\ \\ \\ \purple{ \boxed { \implies \frac{1}{1 + {x}^{2} } }}$

6 0

1 year ago

|1.| 85 is 20% of what number???? | 2.| 22 is 11% of what number???

Kazeer [188]

1) 85 is 20% of 425 (you can solve this by dividing 85 and 20%)

2) 22 is 11% of 200 (divide)

3) 60% of 145 is 87 (multiply 60% and 145)

4) he spent $33 (multiply)

5) B.25 (divide)

3 0

3 years ago

Len [333]

Answer:

what should we have to do here

I don't understand

4 0

2 years ago

What is 2×(-5)-(-8)= A. 79 B.-80 C.80 D.83 what is the answer?

saul85 [17]

The answer is -2 when you simplify

6 0

3 years ago

What would x be in 5x=55

BabaBlast [244]

Answer:

5x=55

Divide both sides by 5 to solve for x

5x/5=55/5

x=11

8 0

3 years ago

Read 2 more answers