Dạng 5: Tổng hợp nâng cao hơn

a, $3.5^2-16:2^2$ = 3.25 $-$ 16:4 = 75$-$4 = 71

b, 15.141 + 59.15 = 15.(141+59) = 15.200 = 3000

c, $2^3.17-2^3.14$ = 8.17$-$8.14 = 8.(17$-$14) = 8.3 = 24

d, $20-\left[30-{\left(5-1\right)}^2\right]$ = $20-\left(30-4^2\right)$ = 20$-$(30$-$16) = 20$-$14 = 6

a, $3^5-{10}^2:5^2+2^3.7$ = 243$-$100:25+8.7 = 243$-$4+56 = 239+56 = 295

b, $36.57-36.35+64.62-64.40$ = 36(57$-$35)+64(62$-$40)

= 36.22+64.22 = 22.(36+64) = 22.100 = 2200

a, $600:\{450:\left[450-\left(4.5^3-2^3.5^2\right)\right]\}$

= 600:{450:[450$-$(4.125$-$8.25)]} = 600:{450:[450$-$(500$-$200)]}

= 600:{450:[450$-$300]} = 600:{450:150} = 600:3 = 200

b, $18.\{420:6-\left[150-(68.2-2^3.5)\right]\}$

= 18.{70$-$[150$-$(68.2$-$8.5)]} = 18.{70$-$[150$-$(136$-$40)]}

= 18.{70$-$[150$-$96]} = 18.{70$-$54} = 18.16 = 288

a, A = 27.36+73.99+27.14$-$49.73

= 27(36+14)+73(99$-$49)

= 27.50 + 73.50 = 50(27+73) = 50.100 = 5000

b, B = $\left(4^5.10.5^6+{25}^5.2^8\right):\left(2^8.5^4+5^7.2^5\right)$

Ta có: $4^5={\left(2^2\right)}^5=2^{2.5}=2^{10}$; ${25}^5={\left(5^2\right)}^5=5^{2.5}=5^{10}$

=> $4^5.10.5^6+{25}^5.2^8=2^{10}.2.5.5^6+5^{10}.2^8$ = $2^{11}.5^7+5^{10}.2^8$ = $2^8.2^3.5^7+5^7.5^3.2^8$ = $2^8.5^7.\left(2^3+5^3\right)$

Ta lại có: $2^8.5^4+5^7.2^5=2^5.2^3.5^4+5^4.5^3.2^5$ = $2^5.5^4.\left(2^3+5^3\right)$

Suy ra B = $2^8.5^7.\left(2^3+5^3\right)$:[$2^5.5^4.\left(2^3+5^3\right)$] = $2^8.5^7:\left(2^5.5^4\right)$

= $\left(2^8:2^5\right).\left(5^7:5^4\right)$ = $2^3.5^3={\left(2.5\right)}^3={10}^3=1000$

a, $70-5(x-3)=45$

$\iff 5.\left(x-3\right)=70-45$

$\iff 5\left(x-3\right)=35$

$\iff x-3=35:5\iff x-3=7$

$\iff x=10$

b, $10+2x=4^5:4^3$

$\iff 10+2x=4^2\iff 10+2x=16$

$\iff 2x=4\iff x=2$

c, $60-3\left(x-2\right)=51$

$\iff 3\left(x-2\right)=60-51$

$\iff 3\left(x-2\right)=9$

$\iff x-2=3\iff x=5$

d, $4x-20=2^5:2^3$

$\iff 4x-20=2^2\iff 4x-20=4$

$\iff 4x=24\iff x=6$

a, $1170:\left(x-13\right)=339.3.{10}^2$

1170:(x$-$13) = 339$-$3.100

1170:(x$-$13) = 39

x$-$13 = 1170:39 = 30 

x = 30+13 = 43

b, $3\left(x-9\right)+15=12+243:3^2$

3(x$-$9) + 15 = 12+243:9

3(x$-$9)+15 = 12+27

3(x$-$9)+15 = 39

3(x$-$9) = 24

x $-$ 9 = 8

x = 17

a, 149$-$(35:x+3).17 = 13

<=> (35:x+3).17 = 136

<=> 35:x+3 = 8

<=> 35:x = 5

<=> x = 7

b, $\overline{1x32}+\overline{7x8}+\overline{4x}=\overline{200x}$

<=> (1000+100x+32)+(700+10x+8)+(40+x) = 2000 + x

<=> (1000+32+700+8+40)+(100x+10x+x) = 2000+x

<=> 1780+111x = 2000+x

<=> 110x = 220

<=> x = 2

a, ${\left(3x-2\right)}^3=2.32$

${\left(3x-2\right)}^3=64$

${\left(3x-2\right)}^3=4^3$

$3x-2=4\iff x=2$

b, $5^{x+1}-5^x=500$

$5^x.5-5^x=500$

$5^x.(5-1)=500$

$5^x.4=500$

$5^x=125=5^3$

$x=3$

Thay  = 3; b = 5; c = 7; d = 1 vào biểu thức P, ta được:

$P=3.3.5^2-\frac{3^3}{1}+7$ = 9.25$-$27+7 = 225$-$27+7 = 198+7=205

Vậy P = 205 khi  = 3; b = 5; c = 7; d = 1

Tích mới hơn tích cũ là: 850 – 476 = 374

Tích mới hơn tích cũ là 374 vì được thêm 22 lần thừa số kia.

Thừa số thứ hai là: 374 : 22 = 17

Thừa số thứ nhất là: 476 : 17 = 28.

Vậy hai số cần tìm là: 17 và 28

Ta có: $A=3+3^2+3^3+...+3^{100}$

=> $3A=3^2+3^3+3^4+...+3^{101}$

=> $3A-A=(3^2+3^3+3^4+...+3^{101})$$-$$(3+3^2+3^3+...+3^{100})$

=> $2A=3^2+3^3+3^4+...+3^{101}$$-3-3^2-3^3-...-3^{100}$

$2A=3^{101}-3$ <=> $2A+3=3^{101}$, mà $2A+3=3^n$

=> n = 101