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✶

P♦tê♥❝✐❛s ❡ ❘❛í③❡s❀ ◆♦t❛çã♦ ❈✐❡♥tí✜❝❛

❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ❛ ❛❞✐çã♦ ❡ ❛ s✉❜tr❛çã♦ ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ❛ ❞✐✈✐sã♦✱ ❛ ♣♦t❡♥❝✐❛çã♦ ❡ ❛ r❛❞✐❝✐❛çã♦ sã♦ ♦♣❡r❛çõ❡s ✐♥✈❡rs❛s ♥❛ ▼❛t❡♠át✐❝❛✱ ❞❡ ❢♦r♠❛ q✉❡ ❛♣❧✐❝❛♥❞♦ ✉♠❛ ❞❡❧❛s ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♥ú♠❡r♦✱ ♣♦❞❡✲s❡ ✈♦❧t❛r ❛♦ ♠❡s♠♦ ♥ú♠❡r♦ ✭q✉❛♥❞♦ ❞❡✜♥✐❞❛s✮✱ ❛♣❧✐❝❛♥❞♦ ❛ ♦♣❡r❛çã♦ ✐♥✈❡rs❛ ❝♦rr❡s♣♦♥❞❡♥t❡ à ♣r✐♠❡✐r❛✳

P♦tê♥❝✐❛çã♦

❙❡❥❛ n ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ❡ ♠❛✐♦r q✉❡ 1✿ ❛ ♣♦tê♥❝✐❛ ❞❡ ❜❛s❡ a ❡ ❡①♣♦❡♥t❡ n é ♦ ♣r♦❞✉t♦ ❞❡ n ❢❛t♦r❡s ✐❣✉❛✐s ❛ a✳

❘❡♣r❡s❡♥t❛♥❞♦ ❛ ♣♦tê♥❝✐❛ ♣❡❧❛ s✐♠❜♦❧♦❣✐❛an _{✱ t❡♠✲s❡ q✉❡✿}

an=a.a.a...a

❡♠ q✉❡ ♥♦ ♣r♦❞✉t♦ ❛❝✐♠❛ ❛♣❛r❡❝❡♠n❢❛t♦r❡s✳

P❡❧❛ ❞❡✜♥✐çã♦ t❡♠♦s q✉❡ a1=a❡ ♣♦r ❝♦♥✈❡♥çã♦a0= 1✳

P♦❞❡♠♦s t❛♠❜é♠ ❞❡✜♥✐r ♣♦tê♥❝✐❛s ♣❛r❛ ❡①♣♦❡♥t❡s ♥❡❣❛t✐✈♦s✿

a−n

= 1

an

❞❡s❞❡ q✉❡a6= 0.

Pr♦♣r✐❡❞❛❞❡s

✶✳ an_am_=an+m_;

✷✳ (ab)n _=an_bn_;

✸✳ an

am =a

n−m_{; a}

6

= 0;

✹✳ a b

n

= an

bn;

✺✳ (an₎m_=anm_;

✻✳ ❙❡ af(x)_=ag(x) _❡a

6

= 0,1 ❡♥tã♦

f(x) =g(x)✳

❊①❡r❝í❝✐♦s

✶✳ ❈❛❧❝✉❧❡ ❛s ♣♦tê♥❝✐❛s ❛❜❛✐①♦✿ ✭❛✮ 25_;

✭❜✮ 2 3

3

; ✭❝✮ (−1)10_;

✭❞✮ 100_;

✭❡✮ 01_;

✭❢✮ 2−3_;

✭❣✮ 1 3

−2

;

✭❤✮ (₋3)2_;

✭✐✮ −32_;

✷✳ ❈❛❧❝✉❧❡ ❛s ♣♦tê♥❝✐❛s ✉t✐❧✐③❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s✿ ✭❛✮ [(−3)2_]−1_;

✭❜✮ 10−3

÷10−6

10−2 ;

✭❝✮ h− 13

3i−2

;

✭❞✮ 105

·102

÷10−1

10−3_÷10 .

✷

✭❛✮ 2x+3₌1 8;

✭❜✮ 8x2

−x_{= 4}x+1_;

✭❝✮ 4x 5 = 1

23;

✭❞✮ 56x₊₃

= 25x−1

3 .

❘❛í③ ❡♥és✐♠❛ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧

P❛r❛ ✉♠ ♥ú♠❡r♦ r❡❛❧a✱ ❛ ❡①♣r❡ssã♦ √nar❡♣r❡s❡♥t❛ ♦ ú♥✐❝♦ ♥ú♠❡r♦ r❡❛❧xq✉❡ ✈❡r✐✜❝❛xn =a❡ t❡♠ ♦ ♠❡s♠♦ s✐♥❛❧ q✉❡

❛ ✭q✉❛♥❞♦ ❡①✐st❡✮✳ ◗✉❛♥❞♦né ♦♠✐t✐❞♦✱ s✐❣♥✐✜❝❛ q✉❡n= 2❡ ♦ sí♠❜♦❧♦ ❞❡ r❛❞✐❝❛❧ r❡❢❡r❡✲s❡ à r❛✐③ q✉❛❞r❛❞❛✳ ❆x❝❤❛♠❛✲s❡

❛ r❛✐③✱ní♥❞✐❝❡✱ ❛ar❛❞✐❝❛♥❞♦ ❡ ❛√ r❛❞✐❝❛❧✳

❖ ♥ú♠❡r♦ √n_at❛♠❜é♠ ♣♦❞❡ s❡r ❡①❝r✐t♦ ♥❛ ❢♦r♠❛✿

n

√ a=an1

♦ q✉❡ ❞❡✜♥❡ ♣♦tê♥❝✐❛s ❝♦♠ ❡①♣♦❡♥t❡ ❢r❛❝✐♦♥ár✐♦✳ ❆ss✐♠ t❡♠♦s✿

n

√

am_=am n.

Pr♦♣r✐❡❞❛❞❡s

✶✳ √na√n

b= √n

ab;

✷✳ √n_a n

√

b =

n

pa b;

✸✳ (√na)m= √n_am❀

✹✳ np√

amp₌ √n

am_=am n;

✺✳ pn m√a= nm√a.

❊①❡r❝í❝✐♦s

✶✳ ❙✐♠♣❧✐✜q✉❡✿ ✭❛✮ √3

25

;

✭❜✮ 2p√2;

✭❝✮ −√3

−8 + 1614− −1

2

−2

+ 843;

✭❞✮ 5(16)0_{+ 3(16)}3

4 + 4(16)− 1 2;

✭❡✮ q3 p√

8;

✭❢✮ q4 √a 3

√_a

6

.

✷✳ ❘❡s♦❧✈❛ ❛s ❡q✉❛çõ❡s ❡①♣♦♥❡♥❝✐❛✐s✿ ✭❛✮ 3x₌ 1

729;

✭❜✮ 5√4

2x_{= 160;}

✭❝✮ 5 x

√

32 = 2; ✭❞✮ 9x−1_{= 3.}

❘❛❝✐♦♥❛❧✐③❛çã♦ ❞❡ ❞❡♥♦♠✐♥❛❞♦r❡s

5

√

3.

❊❧❛ ♣♦ss✉✐ ❡♠ s❡✉ ❞❡♥♦♠✐♥❛❞♦r ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧✳ ❱❛♠♦s ❛❣♦r❛ ♠✉❧t✐♣❧✐❝❛r ♦ ♥✉♠❡r❛❞♦r ❡ ♦ ❞❡♥♦♠✐♥❛❞♦r ❞❡st❛ ❢r❛çã♦ ♣♦r√3✱ ♦❜t❡♥❞♦ ✉♠❛ ❢r❛çã♦ ❡q✉✐✈❛❧❡♥t❡✿

5√3 3 .

❖❜s❡r✈❡ q✉❡ ❛ ❢r❛çã♦ ❡q✉✐✈❛❧❡♥t❡ ♣♦ss✉✐ ✉♠ ❞❡♥♦♠✐♥❛❞♦r r❛❝✐♦♥❛❧✳ ❆ ❡ss❛ tr❛♥s❢♦r♠❛çã♦✱ ❞❛♠♦s ♦ ♥♦♠❡ ❞❡ r❛❝✐♦♥❛❧✐③❛✲ çã♦ ❞❡ ❞❡♥♦♠✐♥❞♦r❡s✳ ❆ r❛❝✐♦♥❛❧✐③❛çã♦ ❞❡ ❞❡♥♦♠✐♥❛❞♦r❡s ❝♦♥s✐st❡✱ ♣♦rt❛♥t♦✱ ♥❛ ♦❜t❡♥çã♦ ❞❡ ✉♠ ❢r❛çã♦ ❝♦♠ ❞❡♥♦♠✐♥❛❞♦r r❛❝✐♦♥❛❧✱ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ❛♥t❡r✐♦r✱ q✉❡ ♣♦ss✉í❛ ✉♠ ♦✉ ♠❛✐s r❛❞✐❝❛✐s ❡♠ s❡✉ ❞❡♥♦♠✐♥❛❞♦r✳ P❛r❛ r❛❝✐♦♥❛❧✐③❛r ♦ ❞❡♥♦✲ ♠✐♥❛❞♦r ❞❡ ✉♠❛ ❢r❛çã♦ ❞❡✈❡♠♦s ♠✉❧t✐♣❧✐❝❛r ♦s t❡r♠♦s ❞❡st❛ ❢r❛çã♦ ♣♦r ✉♠❛ ❡①♣r❡ssã♦ ❝♦♠ r❛❞✐❝❛❧✱ ❞❡♥♦♠✐♥❛❞♦ ❢❛t♦r r❛❝✐♦♥❛❧✐③❛♥t❡✱ ❞❡ ♠♦❞♦ ❛ ♦❜t❡r ✉♠❛ ♥♦✈❛ ❢r❛çã♦ ❡q✉✐✈❛❧❡♥t❡ ❝♦♠ ❞❡♥♦♠✐♥❛❞♦r s❡♠ r❛❞✐❝❛❧✳

❖s ♣r✐♥❝✐♣❛✐s ❝❛s♦s ❞❡ r❛❝✐♦♥❛❧✐③❛çã♦ sã♦✿

✶✳ ❘❛✐③ q✉❛❞r❛❞❛ √a♥♦ ❞❡♥♦♠✐♥❛❞♦r✿ ♠✉❧t✐♣❧✐❝❛♠♦s ♥✉♠❡r❛❞♦r ❡ ❞❡♥♦♠✐♥❛❞♦r ♣♦r√a✳ m

√ a =

m√a √

a√a = m√a

✸

✷✳ ❘❛✐③ ❡♥és✐♠❛ √n

am_{♥♦ ❞❡♥♦♠✐♥❛❞♦r✿ ♠✉❧t✐♣❧✐❝❛♠♦s ♥✉♠❡r❛❞♦r ❡ ❞❡♥♦♠✐♥❛❞♦r ♣♦r} √n

an₋m_✳

m

n

√ am =

m√n

an₋m

n

√ am√n

an−m =

m√n

an₋m

a .

✸✳ ❖ ❞❡♥♦♠✐♥❛❞♦r ❝♦♥té♠ s♦♠❛ ✭♦✉ ❞✐❢❡r❡♥ç❛✮ ❝♦♠ r❛✐③ q✉❛❞r❛❞❛✿ ♠✉❧t✐♣❧✐❝❛♠♦s ♥✉♠❡r❛❞♦r ❡ ❞❡♥♦♠✐♥❛❞♦r ♣❡❧♦ ❝❤❛♠❛❞♦ ❝♦♥❥✉❣❛❞♦ ❞♦ ❞❡♥♦♠✐♥❞❛♦r✳ ❖ ❝♦♥❥✉❣❛❞♦ ❞❡ b_±√a=b_∓√a✳

m b+√a=

m(b−√a) (b+√a)(b−√a)=

m(b−√a)

b2₋√_a .

❊①❡r❝í❝✐♦s

❘❛❝✐♦♥❛❧✐③❡ ♦s ❞❡♥♦♠✐♥❛❞♦r❡s ❛❜❛✐①♦✿ ✶✳ 2

√

5;

✷✳ 9

√

7;

✸✳ 2

5

√

34;

✹✳ 1

3

√

52;

✺✳ 2 1+√5;

✻✳ 1 2+√2;

✼✳ 3 2−√7;

◆♦t❛çã♦ ❈✐❡♥tí✜❝❛

◆♦t❛çã♦ ❝✐❡♥tí✜❝❛✱ é t❛♠❜é♠ ❞❡♥♦♠✐♥❛❞❛ ♣♦r ♣❛❞rã♦ ♦✉ ♥♦t❛çã♦ ❡♠ ❢♦r♠❛ ❡①♣♦♥❡♥❝✐❛❧✱ é ✉♠❛ ❢♦r♠❛ ❞❡ ❡s❝r❡✈❡r ♥ú♠❡r♦s q✉❡ ❛❝♦♠♦❞❛ ✈❛❧♦r❡s ❞❡♠❛s✐❛❞❛♠❡♥t❡ ❣r❛♥❞❡s✱ ❝♦♠♦ 100000000000✱ ♦✉ ♣❡q✉❡♥♦s✱ ❝♦♠♦ 0,00000000001✱ ♣❛r❛

s❡r❡♠ ❝♦♥✈❡♥✐❡♥t❡♠❡♥t❡ ❡s❝r✐t♦s ❞❡ ❢♦r♠❛ ♣❛❞r♦♥✐③❛❞❛✳ ❖ ✉s♦ ❞❡st❛ ♥♦t❛çã♦ ❡stá ❜❛s❡❛❞♦ ♥❛s ♣♦tê♥❝✐❛s ❞❡ 10✭♦s ❝❛s♦s ❡①❡♠♣❧✐✜❝❛❞♦s ❛❝✐♠❛✱ ❡♠ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛✱ ✜❝❛r✐❛♠✿ 1×1011 _{❡ 1×10}−11_{✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✮✳ ❯♠ ♥ú♠❡r♦ ❡s❝r✐t♦ ❡♠}

♥♦t❛ã☛♦ ❝✐❡♥tí✜❝❛ s❡❣✉❡ ♦ s❡❣✉✐♥t❡ ♠♦❞❡❧♦✿

m×10e.

❖ ♥ú♠❡r♦ mé ❞❡♥♦♠✐♥❛❞♦ ♠❛♥t✐ss❛ ❡e❛ ♦r❞❡♠ ❞❡ ❣r❛♥❞❡③❛✳ ❆ ♠❛♥t✐ss❛✱ ❡♠ ♠ó❞✉❧♦✱ ❞❡✈❡ s❡r ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛1❡ ♠❡♥♦r q✉❡10✳ ❉❡ss❡ ♠♦❞♦ ❝❛❞❛ ♥ú♠❡r♦ é r❡♣r❡s❡♥t❛❞♦ ❞❡ ✉♠❛ ú♥✐❝❛ ♠❛♥❡✐r❛ ♥❛ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛✳ ❖❜s❡r✈❡ ♦s ❡①❡♠♣❧♦s ❞❡ ❝♦♠♦ ♥ú♠❡r♦s ❣r❛♥❞❡s ❡ ♣❡q✉❡♥♦s ♣♦❞❡♠ s❡r ❡s❝r✐t♦s ♥❛ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛✿

• 600000 = 6×105_;

• 30000000 = 3×107_;

• 500000000000000 = 5×1014_;

• 7000000000000000000000000000000000 = 7×1033_;

• 0,0004 = 4×10−4_;

• 0,00000001 = 1×10−8_;

• 0,0000000000000006 = 6×10−16_;

• 0,0000000000000000000000000000000000000000000000008 = 8×10−49_.

❖❜s❡r✈❡ ❛ tr❛♥s❢♦r♠❛çã♦ ♣❛ss♦ ❛ ♣❛ss♦✿

0,0000000475 = 0,000000475_×10−1 ₌

0,00000475×10−2 ₌

0,0000475_×10−3 ₌

0,000475_×10−4 ₌

0,00475×10−5 ₌

0,0475_×10−6 ₌

0,475×10−7 ₌

✹

❆ r❡♣r❡s❡♥t❛çã♦ ❞❡❝✐♠❛❧ ❞❡ss❡s ♥ú♠❡r♦s tr❛③ ♣♦✉❝♦ s✐❣♥✐✜❝❛❞♦ ♣rát✐❝♦✳ P♦❞❡✲s❡ ❛té ♣❡♥s❛r q✉❡ ❡ss❡s ✈❛❧♦r❡s sã♦ ♣♦✉❝♦ r❡❧❡✈❛♥t❡s ❡ ❞❡ ✉s♦ q✉❛s❡ ✐♥❡①✐st❡♥t❡ ♥❛ ✈✐❞❛ ❝♦t✐❞✐❛♥❛✳ P♦ré♠✱ ❡♠ ár❡❛s ❝♦♠♦ ❛ ❢ís✐❝❛ ❡ ❛ q✉í♠✐❝❛✱ ❡ss❡s ✈❛❧♦r❡s sã♦ ❢r❡q✉❡♥t❡s✳ P♦r ❡①❡♠♣❧♦✱ ❛ ♠❛✐♦r ❞✐stâ♥❝✐❛ ♦❜s❡r✈á✈❡❧ ❞♦ ✉♥✐✈❡rs♦ ♠❡❞❡ ❝❡r❝❛ ❞❡ 740000000000000000000000000 m✱ ❡

❛ ♠❛ss❛ ❞❡ ✉♠ ♣rót♦♥ é ❛♣r♦①✐♠❛❞❛♠❡♥t❡ 0,00000000000000000000000000167 kg✳ P❛r❛ ✈❛❧♦r❡s ❝♦♠♦ ❡ss❡s✱ ❛ ♥♦t❛çã♦

❝✐❡♥tí✜❝❛ é ♠❛✐s ❛❞❡q✉❛❞❛✱ ♣♦✐s ❛♣r❡s❡♥t❛ ❛ ✈❛♥t❛❣❡♠ ❞❡ ♣♦❞❡r r❡♣r❡s❡♥t❛r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s✱ q✉❡ sã♦ ♦s ❛❧❣❛r✐s♠♦s q✉❡ ♦s ❛♣❛r❡❧❤♦s ❞❡ ♠❡❞✐❞❛ sã♦ ❝❛♣❛③❡s ❞❡ ❞❡t❡r♠✐♥❛r ❝♦♠ ♣r❡❝✐sã♦✳ P♦r ❡①❡♠♣❧♦✱ ❛ ❞✐stâ♥❝✐❛ ♦❜s❡r✈á✈❡❧ ❞♦ ✉♥✐✈❡rs♦✱ ❞♦ ♠♦❞♦ q✉❡ ❡stá ❡s❝r✐t♦✱ s✉❣❡r❡ ❛ ♣r❡❝✐sã♦ ❞❡ ✷✼ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s✳ ▼❛s ✐ss♦ ♣♦❞❡ ♥ã♦ s❡r ✈❡r❞❛❞❡ ✭é ♣♦✉❝♦ ♣r♦✈á✈❡❧ ✷✺ ③❡r♦s s❡❣✉✐❞♦s ♥✉♠❛ ❛❢❡r✐çã♦✮✳

◆♦t❛çã♦ ❝✐❡♥tí✜❝❛ é ✉♠❛ ❢♦r♠❛ ♠✉✐t♦ ❝♦♥✈❡♥✐❡♥t❡ ♣❛r❛ ❡s❝r❡✈❡r ♣❡q✉❡♥♦s ♦✉ ❣r❛♥❞❡s ♥ú♠❡r♦s ❡ ❢❛③❡r ❝á❧❝✉❧♦s ❝♦♠ ❡❧❡s✳ ❚❛♠❜é♠ tr❛♥s♠✐t❡ r❛♣✐❞❛♠❡♥t❡ ❞✉❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ✉♠❛ ♠❡❞✐❞❛ q✉❡ sã♦ út❡✐s ♣❛r❛ ♦s ❝✐❡♥t✐st❛s✱ ❛❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s ❡ ♦r❞❡♠ ❞❡ ❣r❛♥❞❡③❛✳ ❊s❝r✐t❛ ❡♠ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛ ♣❡r♠✐t❡ ❛ ✉♠❛ ♣❡ss♦❛ ❡❧✐♠✐♥❛r ③❡r♦s ♥❛ ❢r❡♥t❡ ♦✉ ❞❡ trás ❞♦s ❞í❣✐t♦s s✐❣♥✐✜❝❛t✐✈♦s✳

❆❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s

❆❧❣❛r✐s♠♦s s✐❣♥✐✜❝❛t✐✈♦s sã♦ ♦s ❛❧❣❛r✐s♠♦s q✉❡ ♦s ❛♣❛r❡❧❤♦s ❞❡ ♠❡❞✐❞❛ sã♦ ❝❛♣❛③❡s ❞❡ ❞❡t❡r♠✐♥❛r ❝♦♠ ♣r❡❝✐sã♦✳ ❯♠❛ ✈❛♥t❛❣❡♠ ❞❛ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛ é q✉❡ ❡❧❛ r❡❞✉③ ❛ ❛♠❜✐❣✉✐❞❛❞❡ ❞♦ ♥ú♠❡r♦ ❞❡ ❞í❣✐t♦s s✐❣♥✐✜❝❛t✐✈♦s✳ ❚♦❞♦s ♦s ❞í❣✐t♦s ❡♠ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛ ♥♦r♠❛❧✐③❛❞❛ sã♦ s✐❣♥✐✜❝❛t✐✈♦s ♣♦r ❝♦♥✈❡♥çã♦✳ ▼❛s✱ ❡♠ ♥♦t❛çã♦ ❞❡❝✐♠❛❧ q✉❛❧q✉❡r ③❡r♦ ♦✉ ✉♠❛ sér✐❡ ❞❡ ③❡r♦s ❛♦ ❧❛❞♦ ❞♦ ♣♦♥t♦ ❞❡❝✐♠❛❧ sã♦ ❛♠❜í❣✉♦s✱ ❡ ♣♦❞❡♠ ♦✉ ♥ã♦ ✐♥❞✐❝❛r ♥ú♠❡r♦s s✐❣♥✐✜❝❛t✐✈♦s ✭q✉❛♥❞♦ ❡❧❡s ❞❡✈❡♠ s❡r s✉❜❧✐♥❤❛❞♦s ♣❛r❛ ❞❡✐①❛r ❡①♣❧✐❝✐t♦ q✉❡ ❡❧❡s sã♦ ③❡r♦s s✐❣♥✐✜❝❛t✐✈♦s✮✳ ❊♠ ✉♠❛ ♥♦t❛çã♦ ❞❡❝✐♠❛❧✱ ③❡r♦s ❛♦ ❧❛❞♦ ❞♦ ♣♦♥t♦ ❞❡❝✐♠❛❧ ♥ã♦ sã♦✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ✉♠ ♥ú♠❡r♦ s✐❣♥✐✜❝❛t✐✈♦✳ ❖✉ s❡❥❛✱ ❡❧❡s ♣♦❞❡♠ ❡st❛r ❛❧✐ ❛♣❡♥❛s ♣❛r❛ ♠♦str❛r ♦♥❞❡ s❡ ❧♦❝❛❧✐③❛ ♦ ♣♦♥t♦ ❞❡❝✐♠❛❧✳ ❊♠ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛✱ ❝♦♥t✉❞♦✱ ❡ss❛ ❛♠❜✐❣✉✐❞❛❞❡ é r❡s♦❧✈✐❞❛✱ ♣♦rq✉❡ ♦s ③❡r♦s ♠♦str❛❞♦s sã♦ ❝♦♥s✐❞❡r❛❞♦s s✐❣♥✐✜❝❛t✐✈♦s ♣♦r ❝♦♥✈❡♥çã♦✳

➱ ❤❛❜✐t✉❛❧ ❡♠ ♠❡❞✐çõ❡s ❝✐❡♥tí✜❝❛s r❡❣✐str❛r t♦❞♦s ♦s ❞í❣✐t♦s s✐❣♥✐✜❝❛t✐✈♦s ❛ ♣❛rt✐r ❞❛s ♠❡❞✐çõ❡s✱ ❡ s✉♣♦r ✉♠ ❞í❣✐t♦ ❛❞✐❝✐♦♥❛❧✱ s❡ ❤♦✉✈❡r ❛❧❣✉♠❛ ✐♥❢♦r♠❛çã♦ ❞✐s♣♦♥í✈❡❧ ♣❛r❛ ♦ ♦❜s❡r✈❛❞♦r q✉❡ ♣❡r♠✐t❛ ❢❛③❡r ✉♠❛ s✉♣♦s✐çã♦✳ ❖ ♥ú♠❡r♦ r❡s✉❧t❛♥t❡ é ❝♦♥s✐❞❡r❛❞♦ ♠❛✐s ✈❛❧✐♦s♦ ❞♦ q✉❡ s❡r✐❛ s❡♠ ❡ss❡ ❞í❣✐t♦ ❡①tr❛✱ ❡ é ❝♦♥s✐❞❡r❛❞♦ ✉♠ ❞í❣✐t♦ s✐❣♥✐✜❝❛t✐✈♦✱ ♣♦✐s ❝♦♥té♠ ❛❧❣✉♠❛s ✐♥❢♦r♠❛çõ❡s q✉❡ ❝♦♥❞✉③❡♠ ❛ ✉♠❛ ♠❛✐♦r ♣r❡❝✐sã♦ ♥❛s ♠❡❞✐çõ❡s ❡ ♥❛ ❛❣r❡❣❛çã♦ ❞❛s ♠❡❞✐çõ❡s ✭❛❞✐❝✐♦♥á✲❧♦s ♦✉ ♠✉❧t✐♣❧✐❝á✲❧♦s✮✳

❖r❞❡♠ ❞❡ ❣r❛♥❞❡③❛

❆ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛ ♣❡r♠✐t❡ t❛♠❜é♠ ♠❛✐s s✐♠♣❧❡s ❝♦♠♣❛r❛çõ❡s ❡♥tr❡ ♦r❞❡♥s ❞❡ ❣r❛♥❞❡③❛✳ ❆ ♠❛ss❛ ❞❡ ✉♠ ♣rót♦♥ é 0.0000000000000000000000000016726kg✳ ❙❡ ✐st♦ é ❡s❝r✐t♦ ❝♦♠♦1.6726×10−27 _{kg✱ é ♠❛✐s ❢á❝✐❧ ❝♦♠♣❛r❛r ❡ss❛ ♠❛ss❛ ❝♦♠}

❛ ❞♦ ❡❧étr♦♥✱ q✉❡ é9.1093822×10−31_{kg✳ ❆ ♦r❞❡♠ ❞❡ ❣r❛♥❞❡③❛ ❞❛ r❡❧❛çã♦ ❡♥tr❡ ❛s ♠❛ss❛s ♣♦❞❡ s❡r ♦❜t✐❞❛s ❛ ♣❛rt✐r ❞♦s}

❡①♣♦❡♥t❡s ❡♠ ✈❡③ ❞❡ t❡r ❞❡ ❝♦♥t❛r ♦s ③❡r♦s à ❡sq✉❡r❞❛✱ t❛r❡❢❛ ♣r♦♣❡♥s❛ ❛ ❡rr♦s✳ ◆❡ss❡ ❝❛s♦✱−27é ♠❛✐♦r ❞♦ q✉❡ −31❡✱ ♣♦rt❛♥t♦✱ ♦ ♣rót♦♥ é ❛♣r♦①✐♠❛❞❛♠❡♥t❡ q✉❛tr♦ ♦r❞❡♥s ❞❡ ❣r❛♥❞❡③❛ ✭❝❡r❝❛ ❞❡10000✈❡③❡s✮ ♠❛✐s ♠❛❝✐ç♦ q✉❡ ♦ ❡❧étr♦♥✳

❖♣❡r❛çõ❡s

• ❆❞✐çã♦ ❡ s✉❜tr❛çã♦✿ P❛r❛ s♦♠❛r ♦✉ s✉❜tr❛✐r ❞♦✐s ♥ú♠❡r♦s ❡♠ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛✱ é ♥❡❝❡ssár✐♦ q✉❡ ♦s ❡①♣♦❡♥t❡s

s❡❥❛♠ ♦ ♠❡s♠♦✳ ❖✉ s❡❥❛✱ ✉♠ ❞♦s ✈❛❧♦r❡s ❞❡✈❡ s❡r tr❛♥s❢♦r♠❛❞♦ ♣❛r❛ q✉❡ s❡✉ ❡①♣♦❡♥t❡ s❡❥❛ ✐❣✉❛❧ ❛♦ ❞♦ ♦✉tr♦✳ ❆ tr❛♥s❢♦r♠❛çã♦ s❡❣✉❡ ♦ ♠❡s♠♦ ♣r✐♥❝í♣✐♦ q✉❡ ✉s❛♠♦s ♣❛r❛ ❡s❝r❡✈❡r ♦ ♥ú♠❡r♦ ♥❛ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛✳ ❖ r❡s✉❧t❛❞♦ ♣♦ss✐✈❡❧♠❡♥t❡ ♥ã♦ ❡st❛rá ♥❛ ❢♦r♠❛ ♣❛❞r♦♥✐③❛❞❛✱ s❡♥❞♦ ❝♦♥✈❡rt✐❞♦ ♣♦st❡r✐♦r♠❡♥t❡✳

❊①❡♠♣❧♦s ✶✳ ✶✳ 4,2×107_{+ 3,5×10}5_{= 4,2×10}7_{+ 0,035×10}7_{= 4,235×10}7_;

✷✳ 6,32_×109

−6,25_×109_{= 0,07}

×109_{= 7}

×107_.

• ▼✉❧t✐♣❧✐❝❛çã♦✿ ▼✉❧t✐♣❧✐❝❛♠♦s ❛s ♠❛♥t✐ss❛s ❡ s♦♠❛♠♦s ♦s ❡①♣♦❡♥t❡s ❞❡ ❝❛❞❛ ✈❛❧♦r✳ ❖ r❡s✉❧t❛❞♦ ♣♦ss✐✈❡❧♠❡♥t❡ ♥ã♦

s❡rá ♣❛❞r♦♥✐③❛❞♦✱ ♠❛s ♣♦❞❡ s❡r ❝♦♥✈❡rt✐❞♦✳

❊①❡♠♣❧♦s ✷✳ ✶✳ (6,5×108_)×(3,2×105_{) = (6,5×3,5)×10}8+5_{= 20,8×10}13_{= 2,08×10}14_;

✷✳ (4_×106₎

×(1,6_×10−15_{) = (4}

×1,6)_×106−15_{= 6,4}

×10−9_.

• ❉✐✈✐sã♦ ❉✐✈✐❞✐♠♦s ❛s ♠❛♥t✐ss❛s ❡ s✉❜tr❛í♠♦s ♦s ❡①♣♦❡♥t❡s ❞❡ ❝❛❞❛ ✈❛❧♦r✳ ❖ r❡s✉❧t❛❞♦ ♣♦ss✐✈❡❧♠❡♥t❡ ♥ã♦ s❡rá

♣❛❞r♦♥✐③❛❞♦✱ ♠❛s ♣♦❞❡ s❡r ❝♦♥✈❡rt✐❞♦✳

❊①❡♠♣❧♦s ✸✳ ✶✳ (8×1017)÷(2×109) = (8÷2)×1017−9_{= 4}

×108 ✷✳ (2,4_×10−7₎

÷(6,2_×10−11_{) = (2,4}

÷6,2)_×10−7−(−11)_{= 0,3871}

×104_{= 3,871}

×103_.

• ❊①♣♦♥❡♥❝✐❛çã♦✿ ❆ ♠❛♥t✐ss❛ é ❡❧❡✈❛❞❛ ❛♦ ❡①♣♦❡♥t❡ ❡①t❡r♥♦ ❡ ♦ ❝♦♥❣r✉❡♥t❡ ❞❛ ❜❛s❡ ❞❡③ é ♠✉❧t✐♣❧✐❝❛❞♦ ♣❡❧♦ ❡①♣♦❡♥t❡

✺

❊①❡♠♣❧♦ ✶✳ (2×106₎4_{= 2}4_×106×4_{= 16×10}24_{= 1,6×10}25_.

• ❘❛❞✐❝✐❛çã♦✿ ❆♥t❡s ❞❡ ❢❛③❡r ❛ r❛❞✐❝✐❛çã♦ é ♣r❡❝✐s♦ tr❛♥s❢♦r♠❛r ✉♠ ❡①♣♦❡♥t❡ ♣❛r❛ ✉♠ ✈❛❧♦r ♠ú❧t✐♣❧♦ ❞♦ í♥❞✐❝❡✳ ❆♣ós

❢❡✐t♦ ✐ss♦✱ ♦ r❡s✉❧t❛❞♦ é ❛ r❛❞✐❝✐❛çã♦ ❞❛ ♠❛♥t✐ss❛ ♠✉❧t✐♣❧✐❝❛❞❛ ♣♦r ❞❡③ ❡❧❡✈❛❞♦ à r❛③ã♦ ❡♥tr❡ ♦ ❡①♣♦❡♥t❡ ❡ ♦ í♥❞✐❝❡ ❞♦ r❛❞✐❝❛❧✳

❊①❡♠♣❧♦s ✹✳ ✶✳ p

1,6_×1027₌√_16×102_{6 =}√_16×1026

2 = 4×1013;

✷✳ p5

6,7_×1017₌√5

670_×101_{5 =} √5

670_×10155 ≈3,674×105;

❊①❡r❝í❝✐♦s

✶✳ ❊s❝r❡✈❛ ♦s ♥ú♠❡r♦s ✉t✐❧✐③❛♥❞♦ ❛ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛✿

✭❛✮ ✉♠ ♠✐❧❤ã♦❀ ✭❜✮ ✉♠ ❞é❝✐♠♦❀ ✭❝✮ ✉♠ tr✐❧❤ã♦❀ ✭❞✮ ❝❡♠ ♠✐❧❀ ✭❡✮ ✉♠ ♠✐❧és✐♠♦✳

✷✳ ❊s❝r❡✈❛ ♦s ♥ú♠❡r♦s ❛❜❛✐①♦ ❡♠ ♥♦t❛çã♦ ❝✐❡♥t✐✜❝❛✿

✭❛✮ ❆ ❞✐stâ♥❝✐❛ ♠é❞✐❛ ❡♥tr❡ ♦ ❙♦❧ ❡ ❛ ❚❡rr❛ é ❞❡14960000Km❀

✭❜✮ ❆ ♠❛ss❛ ❞♦ ❙♦❧ é ❞❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡1989000000000000000000000000000Kg❀

✭❝✮ ❖ ❞✐â♠❡tr♦ ❞♦ ❙♦❧ é1390000 Km❀

✭❞✮ ❆ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③ é ❞❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡300000000m/s❀

✭❡✮ ❖ r❛✐♦ ❞❡ ✉♠ át♦♠♦ é ❞❡0,00000000005mm✳

✸✳ ■♥❢♦r♠❛çõ❡s ❞❛ r❡✈✐st❛ ❙✉♣❡r ■♥t❡r❡ss❛♥t❡✿ ✏ ❖ ❤♦♠❡♠ ♣r♦❞✉③ ✽ tr✐❧❤õ❡s ❞❡ ❡s♣❡r♠❛t♦③ó✐❞❡s ❞✉r❛♥t❡ ❛ ✈✐❞❛✳ ❊♠ ❝❛❞❛ ❡❥❛❝✉❧❛çã♦✱ sã♦ ❧✐❜❡r❛❞♦s ❡♥tr❡ 250000❡500000✳ ❆ ♠✉❧❤❡r ♥❛s❝❡ ❝♦♠ 400000ó✈✉❧♦s ♥♦s ❞♦✐s ♦✈ár✐♦s✳ ❉❡ss❡s✱ só ✉♥s ✺✵✵ ✈ã♦ ♠❛t✉r❛r✳ ❖s q✉❡ ♥ã♦ ❢♦r❡♠ ❢❡rt✐❧✐③❛❞♦s s❡rã♦ ❡❧✐♠✐♥❛❞♦s ♣❡❧❛ ♠❡♥str✉❛çã♦✳✑ ❊s❝r❡✈❛ ❡♠ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛ ♦s ♥ú♠❡r♦s q✉❡ ❛♣❛r❡❝❡♠ ♥❛ r❡♣♦rt❛❣❡♠ ❛❝✐♠❛✳

✹✳ ✭❯♥❡s♣✮ ❈♦♥s✐❞❡r❡ ♦s três ❝♦♠♣r✐♠❡♥t♦s s❡❣✉✐♥t❡s✿ d1= 0,521km✱d2= 5,21×10−2m ❡d3= 5,21×10−6mm✳

✭❛✮ ❊s❝r❡✈❛ ❡ss❡s ❝♦♠♣r✐♠❡♥t♦s ❡♠ ♦r❞❡♠ ❝r❡s❝❡♥t❡❀ ✭❜✮ ❉❡t❡r♠✐♥❡ ❛ r❛③ã♦d3/d2✳

✺✳ ✭❋❡✐✮ ❆ ♠❛ss❛ ❞♦ s♦❧ é ❝❡r❝❛ ❞❡ 1,99_×1030 _{kg✳ ❆ ♠❛ss❛ ❞♦ át♦♠♦ ❞❡ ❤✐❞r♦❣ê♥✐♦✱ ❝♦♥st✐t✉✐♥t❡ ♣r✐♥❝✐♣❛❧ ❞♦ s♦❧ é}

1,67_×10−27 _{kg✳ ◗✉❛♥t♦s át♦♠♦s ❞❡ ❤✐❞r♦❣ê♥✐♦ ❤á ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ♥♦ s♦❧❄}

✻✳ ✭❯❢♣❡✮ ❖ ✢✉①♦ t♦t❛❧ ❞❡ s❛♥❣✉❡ ♥❛ ❣r❛♥❞❡ ❝✐r❝✉❧❛çã♦✱ t❛♠❜é♠ ❝❤❛♠❛❞♦ ❞❡ ❞é❜✐t♦ ❝❛r❞í❛❝♦✱ ❢❛③ ❝♦♠ q✉❡ ♦ ❝♦r❛çã♦ ❞❡ ✉♠ ❤♦♠❡♠ ❛❞✉❧t♦ s❡❥❛ r❡s♣♦♥sá✈❡❧ ♣❡❧♦ ❜♦♠❜❡❛♠❡♥t♦✱ ❡♠ ♠é❞✐❛✱ ❞❡ 20 ❧✐tr♦s ♣♦r ♠✐♥✉t♦✳ ◗✉❛❧ ❛ ♦r❞❡♠ ❞❡ ❣r❛♥❞❡③❛ ❞♦ ✈♦❧✉♠❡ ❞❡ s❛♥❣✉❡✱ ❡♠ ❧✐tr♦s✱ ❜♦♠❜❡❛❞♦ ♣❡❧♦ ❝♦r❛çã♦ ❡♠ ✉♠ ❞✐❛❄

✼✳ ❆ ♥♦ss❛ ❣❛❧á①✐❛✱ ❛ ❱✐❛ ▲á❝t❡❛✱ ❝♦♥té♠ ❝❡r❝❛ ❞❡400❜✐❧❤õ❡s ❞❡ ❡str❡❧❛s✳ ❙✉♣♦♥❤❛ q✉❡0,05%❞❡ss❛s ❡str❡❧❛s ♣♦ss✉❛♠ ✉♠ s✐st❡♠❛ ♣❧❛♥❡tár✐♦ ♦♥❞❡ ❡①✐st❛ ✉♠ ♣❧❛♥❡t❛ s❡♠❡❧❤❛♥t❡ à ❚❡rr❛✳◗✉❛❧ é ♥ú♠❡r♦ ❞❡ ♣❧❛♥❡t❛s s❡♠❡❧❤❛♥t❡s à ❚❡rr❛ ♥❛ ❱✐❛ ▲á❝t❡❛ ❡♠ ♥♦t❛çã♦ ❝✐❡♥tí✜❝❛❄

✽✳ ❉❛❞♦s ♦s ♥ú♠❡r♦s M = 9,84×1015 _{❡N = 1,23×1016♣♦❞❡♠♦s ❛✜r♠❛r q✉❡✿}

✭❛✮ M < N❀

✭❜✮ M +N= 1,07_×1016_❀

✭❝✮ M > N❀

✭❞✮ M _×N= 1,21_×1031_.

✾✳ ❈❛❧❝✉❧❡ 4,1_×10−5_{+ 3}

×10−4

✶✵✳ ◗✉❛♥t♦ ✈❛❧❡105_{+ [(2}

×10−4₎2

×(6,2_×10−8_)]/(4

×10−15_{) +}p3