https://peda.net/id/a356a4e032b 2020-01-09T10:11:28+02:00 https://peda.net/:static/535/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> https://peda.net/id/b4441bfe32c 2020-01-09T11:45:00+02:00 n=1,2,3,4...<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cleft(%5Cfrac%7Bn%7D%7Bn%2B1%7D%5Cright)%5E%7B1-n%7D" alt="a_n=\left(\frac{n}{n+1}\right)^{1-n}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=n%3D1" alt="n=1"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(%5Cfrac%7B1%7D%7B1%2B1%7D%5Cright)1-1%3D1" alt="=\left(\frac{1}{1+1}\right)1-1=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=n%3D2" alt="n=2"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(%5Cfrac%7B2%7D%7B2%2B1%7D%5Cright)%5E%7B1-2%7D%3D%5Cfrac%7B3%7D%7B2%7D" alt="=\left(\frac{2}{2+1}\right)^{1-2}=\frac{3}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=n%3D3" alt="n=3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(%5Cfrac%7B3%7D%7B3%2B1%7D%5Cright)%5E%7B1-3%7D%3D%5Cfrac%7B16%7D%7B4%7D%3D4" alt="=\left(\frac{3}{3+1}\right)^{1-3}=\frac{16}{4}=4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=n%3D4" alt="n=4"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(%5Cfrac%7B4%7D%7B4%2B1%7D%5Cright)%5E%7B1-4%7D%3D%5Cfrac%7B125%7D%7B64%7D" alt="=\left(\frac{4}{4+1}\right)^{1-4}=\frac{125}{64}"/> 2020-01-09T11:45:00+02:00 https://peda.net/id/0249633c32c 2020-01-09T11:40:02+02:00 <div><img src="https://math-demo.abitti.fi/math.svg?latex=10%5E%7B-10%7Dmm%5E3%3D10%5E%7B-16%7Ddm%5E3" alt="10^{-10}mm^3=10^{-16}dm^3"/></div> <div>vesimolekyylien määrä maljakossa on</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3%5Ccdot10%5E%7B25%7D%5Ccdot10%5E%7B-16%7Ddm%5E3%3D3%5Ccdot10%5E9" alt="3\cdot10^{25}\cdot10^{-16}dm^3=3\cdot10^9"/></div> <div>maljakkoon mahtuu 3 miljardia vesimolekyyliä</div> 2020-01-09T11:40:02+02:00 https://peda.net/id/8c46088032c 2020-01-09T11:29:34+02:00 a)<br/> 0,00123<br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-5%5Ccdot10%5E%7B-4%7D" alt="-5\cdot10^{-4}"/><br/> c)<br/> 537,34<br/> d)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=7%5Ccdot10%5E1%2B5%5Ccdot10%5E0%2B2%5Ccdot10%5E%7B-1%7D%2B3%5Ccdot10%5E%7B-2%7D" alt="7\cdot10^1+5\cdot10^0+2\cdot10^{-1}+3\cdot10^{-2}"/> 2020-01-09T11:29:34+02:00 https://peda.net/id/0db0b89e32c 2020-01-09T11:32:53+02:00 a) <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Cfrac%7B5%7D%7B4%7D%5Cright)%5E%7B-2%7D%2B5%5E%7B-1%7D%3D%5Cfrac%7B16%7D%7B25%7D%2B%5Cfrac%7B1%7D%7B5%7D%3D%5Cfrac%7B21%7D%7B25%7D" alt="\left(\frac{5}{4}\right)^{-2}+5^{-1}=\frac{16}{25}+\frac{1}{5}=\frac{21}{25}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Cfrac%7B1%7D%7B2x%7D%5Cright)%5E%7B-3%7D-%5Cleft(-2x%5Cright)%5E3%3D8x%5E3%2B8x%5E3%3D16x%5E3" alt="\left(\frac{1}{2x}\right)^{-3}-\left(-2x\right)^3=8x^3+8x^3=16x^3"/></div> <div>c)</div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(7x%5Cright)%5E%7B11%7D%5Ccdot%5Cleft(%5Cfrac%7B7%7D%7Bx%7D%5Cright)%5E%7B-9%7D%3D%5Cfrac%7B7%5E%7B11%7Dx%5E%7B11%7D%7D%7B1%7D%5Ccdot%5Cfrac%7Bx%5E9%7D%7B7%5E9%7D%3D%5Cfrac%7B7%5E2x%5E%7B20%7D%7D%7B1%7D%3D49x%5E%7B20%7D" alt="\left(7x\right)^{11}\cdot\left(\frac{7}{x}\right)^{-9}=\frac{7^{11}x^{11}}{1}\cdot\frac{x^9}{7^9}=\frac{7^2x^{20}}{1}=49x^{20}"/> 2020-01-09T11:26:02+02:00 https://peda.net/id/a7c7ee4a32c 2020-01-09T11:16:01+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%5E%7B-4%7D%3D%5Cfrac%7B2%7D%7Bx%5E4%7D" alt="2x^{-4}=\frac{2}{x^4}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3%7D%7B5%7Dx%5E%7B-2%7D%3D%5Cfrac%7B3%7D%7B5x%5E2%7D" alt="\frac{3}{5}x^{-2}=\frac{3}{5x^2}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Cfrac%7B5%7D%7B4x%7D%5Cright)%5E%7B-2%7D%3D%5Cfrac%7B16x%5E2%7D%7B25%7D" alt="\left(\frac{5}{4x}\right)^{-2}=\frac{16x^2}{25}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B2x%5E3%7D%3D2x%5E%7B-3%7D" alt="\frac{1}{2x^3}=2x^{-3}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%7D%7B3x%5E5%7D%3D%5Cfrac%7B2%7D%7B3%7Dx%5E%7B-5%7D" alt="\frac{2}{3x^5}=\frac{2}{3}x^{-5}"/></div> 2020-01-09T11:16:01+02:00 https://peda.net/id/03e7b18e32c 2020-01-09T11:11:26+02:00 a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E3%5Ccdot%20x%5E8%3Dx%5E%7B11%7D" alt="x^3\cdot x^8=x^{11}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4x%5E7%5Ccdot8x%5E%7B14%7D%3D32x%5E%7B21%7D" alt="4x^7\cdot8x^{14}=32x^{21}"/><br/> c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-x%5E6%5Ccdot5x%5E%7B-5%7D%3D-5x" alt="-x^6\cdot5x^{-5}=-5x"/></div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(3x%5Cright)%5E2%5Ccdot%5Cleft(-2x%5Cright)%5E3%3D9x%5E2%5Ccdot%5Cleft(-6x%5E3%5Cright)%3D-54x%5E5" alt="\left(3x\right)^2\cdot\left(-2x\right)^3=9x^2\cdot\left(-6x^3\right)=-54x^5"/></div> 2020-01-09T11:11:26+02:00 https://peda.net/id/4a9fe6d832b 2020-01-09T11:06:16+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(5x%5Cright)%5E2%3D25x%5E2" alt="\left(5x\right)^2=25x^2"/><br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x%5E2%5Cright)%5E4%3Dx%5E8" alt="\left(x^2\right)^4=x^8"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(7x%5E3%5Cright)%5E2%3D49x%5E6" alt="\left(7x^3\right)^2=49x^6"/></div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(-3x%5E5%5Cright)%5E2%3D9x%5E%7B10%7D" alt="\left(-3x^5\right)^2=9x^{10}"/></div> 2020-01-09T11:06:16+02:00 https://peda.net/id/12de91e232b 2020-01-09T10:50:23+02:00 a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3%5E1%2B3%5E0%2B3%5E%7B-1%7D%3D3%2B1%2B%5Cfrac%7B1%7D%7B3%7D%3D4%5C%20%5Cfrac%7B1%7D%7B3%7D" alt="3^1+3^0+3^{-1}=3+1+\frac{1}{3}=4\ \frac{1}{3}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4%5E%7B-2%7D%2B2%5E%7B-1%7D-2%5E1%3D%5Cfrac%7B1%7D%7B16%7D%2B%5Cfrac%7B1%7D%7B2%7D-2%3D%5Cfrac%7B9%7D%7B16%7D-2%3D-%5Cfrac%7B23%7D%7B16%7D" alt="4^{-2}+2^{-1}-2^1=\frac{1}{16}+\frac{1}{2}-2=\frac{9}{16}-2=-\frac{23}{16}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B10%7D-10%5E%7B-2%7D%2B0%7B%2C%7D02%3D0%7B%2C%7D1-%5Cfrac%7B1%7D%7B100%7D%2B0%7B%2C%7D02%3D0%7B%2C%7D11" alt="\frac{1}{10}-10^{-2}+0{,}02=0{,}1-\frac{1}{100}+0{,}02=0{,}11"/></div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-3%5E%7B-2%7D%2B%5Cleft(-6%5Cright)%5E%7B-2%7D%3D-%5Cfrac%7B1%7D%7B9%7D%2B%5Cfrac%7B1%7D%7B36%7D%3D-%5Cfrac%7B4%7D%7B36%7D%2B%5Cfrac%7B1%7D%7B36%7D%3D-%5Cfrac%7B3%7D%7B36%7D%3D-%5Cfrac%7B1%7D%7B12%7D" alt="-3^{-2}+\left(-6\right)^{-2}=-\frac{1}{9}+\frac{1}{36}=-\frac{4}{36}+\frac{1}{36}=-\frac{3}{36}=-\frac{1}{12}"/></div> 2020-01-09T10:50:23+02:00 https://peda.net/id/533dbed032b 2020-01-09T10:45:02+02:00 a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=5%5E3%5Ccdot2%5E3%3D10%5E3%3D1000" alt="5^3\cdot2^3=10^3=1000"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B18%5E4%7D%7B9%5E4%7D%3D%5Cleft(%5Cfrac%7B18%7D%7B9%7D%5Cright)%5E4%3D2%5E4%3D16" alt="\frac{18^4}{9^4}=\left(\frac{18}{9}\right)^4=2^4=16"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Cfrac%7B1%7D%7B3%7D%5Cright)%5E7%5Ccdot3%5E6%3D3%5E%7B-7%7D%5Ccdot3%5E6%3D3%5E%7B-1%7D%3D%5Cfrac%7B1%7D%7B3%7D" alt="\left(\frac{1}{3}\right)^7\cdot3^6=3^{-7}\cdot3^6=3^{-1}=\frac{1}{3}"/></div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%7B%2C%7D5%5Ccdot10%5E3%5Ccdot4%5Ccdot10%5E%7B-4%7D%3D10%5Ccdot10%5E3%5Ccdot10%5E%7B-4%7D%3D10%5E0%3D1" alt="2{,}5\cdot10^3\cdot4\cdot10^{-4}=10\cdot10^3\cdot10^{-4}=10^0=1"/></div> 2020-01-09T10:45:01+02:00 https://peda.net/id/28af396032b 2020-01-09T10:36:41+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=s%3Dkv%5E2" alt="s=kv^2"/>, missä k on vakio<br/> nopeudella 80 km/h jarrutusmatka on 30m, eli <img src="https://math-demo.abitti.fi/math.svg?latex=30%3Dk%5Ccdot80%5E2" alt="30=k\cdot80^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=k%3D%5Cfrac%7B30%7D%7B80%5E2%7D%3D%5Cfrac%7B3%7D%7B640%7D" alt="k=\frac{30}{80^2}=\frac{3}{640}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=s%3D%5Cfrac%7B3%7D%7B640%7Dv%5E2" alt="s=\frac{3}{640}v^2"/><br/> b)<br/> <div>Lasketaan jarrutusmatka nopeuden ollessa 100 km/h</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=s%3D%5Cfrac%7B3%7D%7B640%7D%5Ccdot100%5E2%3D46%7B%2C%7D875m%5Capprox47m" alt="s=\frac{3}{640}\cdot100^2=46{,}875m\approx47m"/></div> <div>nopeudella 80 km/h jarrutusmatkan pituus on 30m</div> <div>matka vähenee <img src="https://math-demo.abitti.fi/math.svg?latex=47m-30m%3D17m" alt="47m-30m=17m"/></div> <div> </div> <div>lasketaan jarrutusmatka nopeuden ollessa 50km/h</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=s%3D%5Cfrac%7B3%7D%7B640%7D%5Ccdot50%5E2%3D11%7B%2C%7D71875%5Capprox12m" alt="s=\frac{3}{640}\cdot50^2=11{,}71875\approx12m"/></div> <div>lasketaan jarrutusmatka nopeuden ollessa 30km/h</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=s%3D%5Cfrac%7B3%7D%7B640%7D%5Ccdot30%5E2%3D4%7B%2C%7D21875%5Capprox4m" alt="s=\frac{3}{640}\cdot30^2=4{,}21875\approx4m"/></div> <div>jarrutusmatka vähenee</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=12m-4m%3D8m" alt="12m-4m=8m"/></div> 2020-01-09T10:36:41+02:00