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cot(81)

Solution

cot (8 1^{\circ})

Solution

\frac{3 2 5 + 5 + 10 5 + 5 + 11 + 4 5}{3 10 + 2 5 + 50 + 10 5 + 11 + 4 5}

Decimal Notation

0.15838 \dots

Solution steps

cot (8 1^{\circ})

Rewrite using trig identities:

\frac{1}{tan ( 8 1 ^{\circ} )}

cot (8 1^{\circ})

Use the basic trigonometric identity:

cot (x) = \frac{1}{tan ( x )}

= \frac{1}{tan ( 8 1 ^{\circ} )}

= \frac{1}{tan ( 8 1 ^{\circ} )}

Rewrite using trig identities:

tan (8 1^{\circ}) = 32 5 + 5 + 10 5 + 5 + 11 + 45

tan (8 1^{\circ})

Rewrite using trig identities:

\frac{1 - cos ( 16 2 ^{\circ} )}{1 + cos ( 16 2 ^{\circ} )}

tan (8 1^{\circ})

Write

tan (8 1^{\circ})

tan (\frac{16 2 ^{\circ}}{2})

= tan (\frac{16 2 ^{\circ}}{2})

Use the Half Angle identity:

tan (\frac{θ}{2}) = \frac{1 - cos ( θ )}{1 + cos ( θ )}

Rewrite using trig identities:

tan^{2} (θ) = \frac{1 - cos ( 2 θ )}{1 + cos ( 2 θ )}

Use the following identity

tan (θ) = \frac{sin ( θ )}{cos ( θ )}

Square both sides

tan^{2} (θ) = \frac{sin ^{2} ( θ )}{cos ^{2} ( θ )}

Rewrite using trig identities:

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

Use the Double Angle identity

cos (2 θ) = 1 - 2 sin^{2} (θ)

Switch sides

2 sin^{2} (θ) - 1 = - cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 - cos (2 θ)

Divide both sides by

2

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

Rewrite using trig identities:

cos^{2} (θ) = \frac{1 + cos ( 2 θ )}{2}

Use the Double Angle identity

cos (2 θ) = 2 cos^{2} (θ) - 1

Switch sides

2 cos^{2} (θ) - 1 = cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 + cos (2 θ)

Divide both sides by

2

cos^{2} (θ) = \frac{1 + cos ( 2 θ )}{2}

tan^{2} (θ) = \frac{\frac{1 - c o s ( 2 θ )}{2}}{\frac{1 + c o s ( 2 θ )}{2}}

Simplify

tan^{2} (θ) = \frac{1 - cos ( 2 θ )}{1 + cos ( 2 θ )}

Substitute

θ

with

\frac{θ}{2}

tan^{2} (\frac{θ}{2}) = \frac{1 - cos ( 2 \cdot \frac{θ}{2} )}{1 + cos ( 2 \cdot \frac{θ}{2} )}

Simplify

tan^{2} (\frac{θ}{2}) = \frac{1 - cos ( θ )}{1 + cos ( θ )}

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

range [0, 9 0^{\circ}] [9 0^{\circ}, 18 0^{\circ}] quadrant I II tan positive negative

tan (\frac{θ}{2}) = \frac{1 - cos ( θ )}{1 + cos ( θ )}

= \frac{1 - cos ( 16 2 ^{\circ} )}{1 + cos ( 16 2 ^{\circ} )}

= \frac{1 - cos ( 16 2 ^{\circ} )}{1 + cos ( 16 2 ^{\circ} )}

Rewrite using trig identities:

cos (16 2^{\circ}) = - \frac{2 5 + 5}{4}

cos (16 2^{\circ})

Rewrite using trig identities:

- cos (1 8^{\circ})

cos (16 2^{\circ})

Use the basic trigonometric identity:

cos (x) = - cos (18 0^{\circ} - x)

= - cos (18 0^{\circ} - 16 2^{\circ})

Simplify

= - cos (1 8^{\circ})

= - cos (1 8^{\circ})

Rewrite using trig identities:

cos (1 8^{\circ}) = \frac{2 5 + 5}{4}

cos (1 8^{\circ})

Rewrite using trig identities:

\frac{1 + cos ( 3 6 ^{\circ} )}{2}

cos (1 8^{\circ})

Write

cos (1 8^{\circ})

cos (\frac{3 6 ^{\circ}}{2})

= cos (\frac{3 6 ^{\circ}}{2})

Use the Half Angle identity:

cos (\frac{θ}{2}) = \frac{1 + cos ( θ )}{2}

Use the Double Angle identity

cos (2 θ) = 2 cos^{2} (θ) - 1

Substitute

θ

with

\frac{θ}{2}

cos (θ) = 2 cos^{2} (\frac{θ}{2}) - 1

Switch sides

2 cos^{2} (\frac{θ}{2}) = 1 + cos (θ)

Divide both sides by

2

cos^{2} (\frac{θ}{2}) = \frac{( 1 + cos ( θ ) )}{2}

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

range [0, 9 0^{\circ}] [9 0^{\circ}, 18 0^{\circ}] [18 0^{\circ}, 27 0^{\circ}] [27 0^{\circ}, 36 0^{\circ}] quadrant I II III I V sin positive positive negative negative cos positive negative negative positive

cos (\frac{θ}{2}) = \frac{( 1 + cos ( θ ) )}{2}

= \frac{1 + cos ( 3 6 ^{\circ} )}{2}

= \frac{1 + cos ( 3 6 ^{\circ} )}{2}

Rewrite using trig identities:

cos (3 6^{\circ}) = \frac{5 + 1}{4}

cos (3 6^{\circ})

Show that:

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

Use the following product to sum identity:

2 sin (x) cos (y) = sin (x + y) - sin (x - y)

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = sin (5 4^{\circ}) - sin (1 8^{\circ})

Show that:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Use the following identity:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

Divide both sides by

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

Substitute

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

\frac{1}{2} = sin (5 4^{\circ}) - sin (1 8^{\circ})

sin (5 4^{\circ}) = cos (9 0^{\circ} - 5 4^{\circ})

\frac{1}{2} = cos (9 0^{\circ} - 5 4^{\circ}) - sin (1 8^{\circ})

\frac{1}{2} = cos (3 6^{\circ}) - sin (1 8^{\circ})

Show that:

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

Use the factorization rule:

a^{2} - b^{2} = (a + b) (a - b)

a = cos (3 6^{\circ}) + sin (1 8^{\circ})

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = ((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) ((cos (3 6^{\circ}) + sin (1 8^{\circ})) - (cos (3 6^{\circ}) - sin (1 8^{\circ})))

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 2 (2 cos (3 6^{\circ}) sin (1 8^{\circ}))

Show that:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Use the following identity:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

Divide both sides by

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

Substitute

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 1

Substitute

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (\frac{1}{2})^{2} = 1

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} = 1

Add

\frac{1}{4}

to both sides

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} + \frac{1}{4} = 1 + \frac{1}{4}

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} = \frac{5}{4}

Take the square root of both sides

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \pm \frac{5}{4}

cos (3 6^{\circ})

cannot be negative

sin (1 8^{\circ})

cannot be negative

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

Add the following equations

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{2}

((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) = (\frac{5}{2} + \frac{1}{2})

Refine

cos (3 6^{\circ}) = \frac{5 + 1}{4}

= \frac{5 + 1}{4}

= \frac{1 + \frac{5 + 1}{4}}{2}

Simplify

\frac{1 + \frac{5 + 1}{4}}{2} : \frac{2 5 + 5}{4}

\frac{1 + \frac{5 + 1}{4}}{2}

\frac{1 + \frac{5 + 1}{4}}{2} = \frac{5 + 5}{8}

\frac{1 + \frac{5 + 1}{4}}{2}

Join

1 + \frac{5 + 1}{4} : \frac{5 + 5}{4}

1 + \frac{5 + 1}{4}

Convert element to fraction:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} + \frac{5 + 1}{4}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 4 + 5 + 1}{4}

1 \cdot 4 + 5 + 1 = 5 + 5

1 \cdot 4 + 5 + 1

Multiply the numbers:

1 \cdot 4 = 4

= 4 + 5 + 1

Add the numbers:

4 + 1 = 5

= 5 + 5

= \frac{5 + 5}{4}

= \frac{\frac{5 + 5}{4}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{5 + 5}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{5 + 5}{8}

= \frac{5 + 5}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{5 + 5}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= \frac{5 + 5}{2 2}

Rationalize

\frac{5 + 5}{2 2} : \frac{2 5 + 5}{4}

\frac{5 + 5}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{5 + 5 2}{2 2 2}

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 5 + 5}{4}

= \frac{2 5 + 5}{4}

= \frac{2 5 + 5}{4}

= - \frac{2 5 + 5}{4}

= \frac{1 - ( - \frac{2 5 + 5}{4} )}{1 - \frac{2 5 + 5}{4}}

Simplify

\frac{1 - ( - \frac{2 5 + 5}{4} )}{1 - \frac{2 5 + 5}{4}} : 32 5 + 5 + 10 5 + 5 + 11 + 45

\frac{1 - ( - \frac{2 5 + 5}{4} )}{1 - \frac{2 5 + 5}{4}}

Apply rule

- (- a) = a

= \frac{1 + \frac{2 5 + 5}{4}}{1 - \frac{2 5 + 5}{4}}

\frac{1 + \frac{2 5 + 5}{4}}{1 - \frac{2 5 + 5}{4}} = \frac{4 + 2 5 + 5}{4 - 2 5 + 5}

\frac{1 + \frac{2 5 + 5}{4}}{1 - \frac{2 5 + 5}{4}}

Join

1 - \frac{2 5 + 5}{4} : \frac{4 - 2 5 + 5}{4}

1 - \frac{2 5 + 5}{4}

Convert element to fraction:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} - \frac{2 5 + 5}{4}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 4 - 2 5 + 5}{4}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4 - 2 5 + 5}{4}

= \frac{1 + \frac{2 5 + 5}{4}}{\frac{4 - 2 5 + 5}{4}}

Join

1 + \frac{2 5 + 5}{4} : \frac{4 + 2 5 + 5}{4}

1 + \frac{2 5 + 5}{4}

Convert element to fraction:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} + \frac{2 5 + 5}{4}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 4 + 2 5 + 5}{4}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4 + 2 5 + 5}{4}

= \frac{\frac{4 + 2 5 + 5}{4}}{\frac{4 - 2 5 + 5}{4}}

Divide fractions:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= \frac{( 4 + 2 5 + 5 ) \cdot 4}{4 ( 4 - 2 5 + 5 )}

Cancel the common factor:

4

= \frac{4 + 2 5 + 5}{4 - 2 5 + 5}

= \frac{4 + 2 5 + 5}{4 - 2 5 + 5}

\frac{4 + 2 5 + 5}{4 - 2 5 + 5} = 32 5 + 5 + 10 5 + 5 + 11 + 45

\frac{4 + 2 5 + 5}{4 - 2 5 + 5}

Multiply by the conjugate

\frac{4 + 2 5 + 5}{4 + 2 5 + 5}

= \frac{( 4 + 2 5 + 5 ) ( 4 + 2 5 + 5 )}{( 4 - 2 5 + 5 ) ( 4 + 2 5 + 5 )}

(4 + 2 5 + 5) (4 + 2 5 + 5) = 82 5 + 5 + 26 + 25

(4 + 2 5 + 5) (4 + 2 5 + 5)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

(4 + 2 5 + 5) (4 + 2 5 + 5) = (4 + 2 5 + 5)^{1 + 1}

= (4 + 2 5 + 5)^{1 + 1}

Add the numbers:

1 + 1 = 2

= (4 + 2 5 + 5)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = 4, b = 2 5 + 5

= 4^{2} + 2 \cdot 42 5 + 5 + (2 5 + 5)^{2}

Simplify

4^{2} + 2 \cdot 42 5 + 5 + (2 5 + 5)^{2} : 82 5 + 5 + 26 + 25

4^{2} + 2 \cdot 42 5 + 5 + (2 5 + 5)^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

2 \cdot 42 5 + 5 = 82 5 + 5

2 \cdot 42 5 + 5

Multiply the numbers:

2 \cdot 4 = 8

= 82 5 + 5

(2 5 + 5)^{2} = 2 (5 + 5)

(2 5 + 5)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= (2)^{2} (5 + 5)^{2}

(2)^{2} : 2

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= 2 (5 + 5)^{2}

(5 + 5)^{2} : 5 + 5

Apply radical rule:

a = a^{\frac{1}{2}}

= ((5 + 5)^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (5 + 5)^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 5 + 5

= 2 (5 + 5)

= 16 + 82 5 + 5 + 2 (5 + 5)

Expand

2 (5 + 5) : 10 + 25

2 (5 + 5)

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = 5, c = 5

= 2 \cdot 5 + 25

Multiply the numbers:

2 \cdot 5 = 10

= 10 + 25

= 16 + 82 5 + 5 + 10 + 25

Add the numbers:

16 + 10 = 26

= 82 5 + 5 + 26 + 25

= 82 5 + 5 + 26 + 25

(4 - 2 5 + 5) (4 + 2 5 + 5) = 6 - 25

(4 - 2 5 + 5) (4 + 2 5 + 5)

2 5 + 5 = 10 + 25

2 5 + 5

Apply radical rule:

a b = a \cdot b

2 5 + 5 = 2 (5 + 5)

= 2 (5 + 5)

Expand

2 (5 + 5) : 10 + 25

2 (5 + 5)

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = 5, c = 5

= 2 \cdot 5 + 25

Multiply the numbers:

2 \cdot 5 = 10

= 10 + 25

= 10 + 25

= (- 10 + 25 + 4) (2 5 + 5 + 4)

2 5 + 5 = 10 + 25

2 5 + 5

Apply radical rule:

a b = a \cdot b

2 5 + 5 = 2 (5 + 5)

= 2 (5 + 5)

Expand

2 (5 + 5) : 10 + 25

2 (5 + 5)

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = 5, c = 5

= 2 \cdot 5 + 25

Multiply the numbers:

2 \cdot 5 = 10

= 10 + 25

= 10 + 25

= (- 10 + 25 + 4) (10 + 25 + 4)

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = 4, b = 10 + 25

= 4^{2} - (10 + 25)^{2}

Simplify

4^{2} - (10 + 25)^{2} : 6 - 25

4^{2} - (10 + 25)^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

(10 + 25)^{2} = 10 + 25

(10 + 25)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= ((10 + 25)^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (10 + 25)^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 10 + 25

= 16 - (10 + 25)

- (10 + 25) : - 10 - 25

- (10 + 25)

Distribute parentheses

= - (10) - (25)

Apply minus-plus rules

+ (- a) = - a

= - 10 - 25

= 16 - 10 - 25

Subtract the numbers:

16 - 10 = 6

= 6 - 25

= 6 - 25

= \frac{8 2 5 + 5 + 26 + 2 5}{6 - 2 5}

Factor

82 5 + 5 + 26 + 25 : 2 (42 5 + 5 + 13 + 5)

82 5 + 5 + 26 + 25

Rewrite as

= 2 \cdot 42 5 + 5 + 2 \cdot 13 + 25

Factor out common term

2

= 2 (42 5 + 5 + 13 + 5)

= \frac{2 ( 4 2 5 + 5 + 13 + 5 )}{6 - 2 5}

Factor

6 - 25 : 2 (3 - 5)

6 - 25

Rewrite as

= 2 \cdot 3 - 25

Factor out common term

2

= 2 (3 - 5)

= \frac{2 ( 4 2 5 + 5 + 13 + 5 )}{2 ( 3 - 5 )}

Divide the numbers:

\frac{2}{2} = 1

= \frac{4 2 5 + 5 + 13 + 5}{( 3 - 5 )}

Remove parentheses:

(a) = a

= \frac{4 2 5 + 5 + 13 + 5}{3 - 5}

Multiply by the conjugate

\frac{3 + 5}{3 + 5}

= \frac{( 4 2 5 + 5 + 13 + 5 ) ( 3 + 5 )}{( 3 - 5 ) ( 3 + 5 )}

(42 5 + 5 + 13 + 5) (3 + 5) = 122 5 + 5 + 410 5 + 5 + 44 + 165

(42 5 + 5 + 13 + 5) (3 + 5)

Distribute parentheses

= 42 5 + 5 \cdot 3 + 42 5 + 5 5 + 13 \cdot 3 + 135 + 5 \cdot 3 + 55

= 4 \cdot 32 5 + 5 + 425 5 + 5 + 13 \cdot 3 + 135 + 35 + 55

Simplify

4 \cdot 32 5 + 5 + 425 5 + 5 + 13 \cdot 3 + 135 + 35 + 55 : 122 5 + 5 + 410 5 + 5 + 44 + 165

4 \cdot 32 5 + 5 + 425 5 + 5 + 13 \cdot 3 + 135 + 35 + 55

Add similar elements:

135 + 35 = 165

= 4 \cdot 32 5 + 5 + 425 5 + 5 + 13 \cdot 3 + 165 + 55

4 \cdot 32 5 + 5 = 122 5 + 5

4 \cdot 32 5 + 5

Multiply the numbers:

4 \cdot 3 = 12

= 122 5 + 5

425 5 + 5 = 410 5 + 5

425 5 + 5

Apply radical rule:

a b = a \cdot b

25 5 + 5 = 2 \cdot 5 (5 + 5)

= 4 2 \cdot 5 (5 + 5)

Multiply the numbers:

2 \cdot 5 = 10

= 4 10 (5 + 5)

Apply radical rule:

assuming

a \geq 0, b \geq 0

10 (5 + 5) = 10 5 + 5

= 410 5 + 5

13 \cdot 3 = 39

13 \cdot 3

Multiply the numbers:

13 \cdot 3 = 39

= 39

55 = 5

55

Apply radical rule:

a a = a

55 = 5

= 5

= 122 5 + 5 + 410 5 + 5 + 39 + 165 + 5

Add the numbers:

39 + 5 = 44

= 122 5 + 5 + 410 5 + 5 + 44 + 165

= 122 5 + 5 + 410 5 + 5 + 44 + 165

(3 - 5) (3 + 5) = 4

(3 - 5) (3 + 5)

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = 3, b = 5

= 3^{2} - (5)^{2}

Simplify

3^{2} - (5)^{2} : 4

3^{2} - (5)^{2}

3^{2} = 9

3^{2}

3^{2} = 9

= 9

(5)^{2} = 5

(5)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (5^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 5^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 5

= 9 - 5

Subtract the numbers:

9 - 5 = 4

= 4

= 4

= \frac{12 2 5 + 5 + 4 10 5 + 5 + 44 + 16 5}{4}

Factor

122 5 + 5 + 410 5 + 5 + 44 + 165 : 4 (32 5 + 5 + 10 5 + 5 + 11 + 45)

122 5 + 5 + 410 5 + 5 + 44 + 165

Rewrite as

= 4 \cdot 32 5 + 5 + 410 5 + 5 + 4 \cdot 11 + 4 \cdot 45

Factor out common term

4

= 4 (32 5 + 5 + 10 5 + 5 + 11 + 45)

= \frac{4 ( 3 2 5 + 5 + 10 5 + 5 + 11 + 4 5 )}{4}

Divide the numbers:

\frac{4}{4} = 1

= 32 5 + 5 + 10 5 + 5 + 11 + 45

= 32 5 + 5 + 10 5 + 5 + 11 + 45

= 32 5 + 5 + 10 5 + 5 + 11 + 45

= \frac{1}{3 2 5 + 5 + 10 5 + 5 + 11 + 4 5}

Simplify

\frac{1}{3 2 5 + 5 + 10 5 + 5 + 11 + 4 5} : \frac{3 2 5 + 5 + 10 5 + 5 + 11 + 4 5}{3 10 + 2 5 + 50 + 10 5 + 11 + 4 5}

\frac{1}{3 2 5 + 5 + 10 5 + 5 + 11 + 4 5}

Multiply by the conjugate

\frac{3 2 5 + 5 + 10 5 + 5 + 11 + 4 5}{3 2 5 + 5 + 10 5 + 5 + 11 + 4 5}

= \frac{1 \cdot 3 2 5 + 5 + 10 5 + 5 + 11 + 4 5}{3 2 5 + 5 + 10 5 + 5 + 11 + 4 5 3 2 5 + 5 + 10 5 + 5 + 11 + 4 5}

1 \cdot 32 5 + 5 + 10 5 + 5 + 11 + 45 = 32 5 + 5 + 10 5 + 5 + 11 + 45

32 5 + 5 + 10 5 + 5 + 11 + 45 32 5 + 5 + 10 5 + 5 + 11 + 45 = 3 10 + 25 + 50 + 105 + 11 + 45

32 5 + 5 + 10 5 + 5 + 11 + 45 32 5 + 5 + 10 5 + 5 + 11 + 45

Apply radical rule:

a a = a

32 5 + 5 + 10 5 + 5 + 11 + 45 32 5 + 5 + 10 5 + 5 + 11 + 45 = 32 5 + 5 + 10 5 + 5 + 11 + 45

= 32 5 + 5 + 10 5 + 5 + 11 + 45

32 5 + 5 = 3 10 + 25

32 5 + 5

Apply radical rule:

a b = a \cdot b

2 5 + 5 = 2 (5 + 5)

= 3 2 (5 + 5)

Expand

2 (5 + 5) : 10 + 25

2 (5 + 5)

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = 5, c = 5

= 2 \cdot 5 + 25

Multiply the numbers:

2 \cdot 5 = 10

= 10 + 25

= 3 10 + 25

10 5 + 5 = 50 + 105

10 5 + 5

Apply radical rule:

a b = a \cdot b

10 5 + 5 = 10 (5 + 5)

= 10 (5 + 5)

Expand

10 (5 + 5) : 50 + 105

10 (5 + 5)

Apply the distributive law:

a (b + c) = ab + a c

a = 10, b = 5, c = 5

= 10 \cdot 5 + 105

Multiply the numbers:

10 \cdot 5 = 50

= 50 + 105

= 50 + 105

= 3 10 + 25 + 50 + 105 + 11 + 45

= \frac{3 2 5 + 5 + 10 5 + 5 + 11 + 4 5}{3 10 + 2 5 + 50 + 10 5 + 11 + 4 5}

= \frac{3 2 5 + 5 + 10 5 + 5 + 11 + 4 5}{3 10 + 2 5 + 50 + 10 5 + 11 + 4 5}

Popular Examples

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