Scientific notation expresses very large or very small numbers as a coefficient multiplied by a power of \(10\). The coefficient must be at least \(1\) but less than \(10\). Positive exponents represent larger numbers, while negative exponents represent numbers between \(0\) and \(1\).
Scientific notation is a method for expressing numbers, and is the most convenient method for very large or very small numbers. Instead of writing the number in decimal form, like this,
\(3{,}285{,}000{,}000{,}000{,}000{,}000{,}000{,}000\)
you can write the number in Scientific Notation like this
\(3.285 \times 10^{24}\)
Same thing for very small numbers. Instead of this
\(0.000000000075\)
you can write the number in Scientific Notation like this
\(7.5 \times 10^{-11}\)
Notes
Problems and Videos
Express in scientific notation
\(\textbf{1)}\) \(125.78\)
The answer is \(1.2578\times10^2\).Move the decimal point \(2\) places to the left.
\(\,\,\,\,\,125.78\rightarrow1.2578\)
Moving the decimal to the left gives a positive exponent.
\(\,\,\,\,\,125.78=1.2578\times10^2\)
The answer is \(1.2578\times10^2\).
\(\,\,\,\,\,125.78\rightarrow1.2578\)
Moving the decimal to the left gives a positive exponent.
\(\,\,\,\,\,125.78=1.2578\times10^2\)
The answer is \(1.2578\times10^2\).
\(\textbf{2)}\) \(0.00000456\)
The answer is \(4.56\times10^{-6}\).Move the decimal point \(6\) places to the right.
\(\,\,\,\,\,0.00000456\rightarrow4.56\)
The original number is between \(0\) and \(1\), so the exponent is negative.
\(\,\,\,\,\,0.00000456=4.56\times10^{-6}\)
The answer is \(4.56\times10^{-6}\).
\(\,\,\,\,\,0.00000456\rightarrow4.56\)
The original number is between \(0\) and \(1\), so the exponent is negative.
\(\,\,\,\,\,0.00000456=4.56\times10^{-6}\)
The answer is \(4.56\times10^{-6}\).
\(\textbf{3)}\) \(1{,}892{,}000{,}000\)
The answer is \(1.892\times10^9\).Move the decimal point \(9\) places to the left.
\(\,\,\,\,\,1{,}892{,}000{,}000\rightarrow1.892\)
Moving the decimal to the left gives a positive exponent.
\(\,\,\,\,\,1{,}892{,}000{,}000=1.892\times10^9\)
The answer is \(1.892\times10^9\).
\(\,\,\,\,\,1{,}892{,}000{,}000\rightarrow1.892\)
Moving the decimal to the left gives a positive exponent.
\(\,\,\,\,\,1{,}892{,}000{,}000=1.892\times10^9\)
The answer is \(1.892\times10^9\).
\(\textbf{4)}\) \(13{,}405\)
The answer is \(1.3405\times10^4\).Move the decimal point \(4\) places to the left.
\(\,\,\,\,\,13{,}405\rightarrow1.3405\)
Moving the decimal to the left gives a positive exponent.
\(\,\,\,\,\,13{,}405=1.3405\times10^4\)
The answer is \(1.3405\times10^4\).
\(\,\,\,\,\,13{,}405\rightarrow1.3405\)
Moving the decimal to the left gives a positive exponent.
\(\,\,\,\,\,13{,}405=1.3405\times10^4\)
The answer is \(1.3405\times10^4\).
\(\textbf{5)}\) \(0.00345\)
The answer is \(3.45\times10^{-3}\).Move the decimal point \(3\) places to the right.
\(\,\,\,\,\,0.00345\rightarrow3.45\)
The original number is between \(0\) and \(1\), so the exponent is negative.
\(\,\,\,\,\,0.00345=3.45\times10^{-3}\)
The answer is \(3.45\times10^{-3}\).
\(\,\,\,\,\,0.00345\rightarrow3.45\)
The original number is between \(0\) and \(1\), so the exponent is negative.
\(\,\,\,\,\,0.00345=3.45\times10^{-3}\)
The answer is \(3.45\times10^{-3}\).
\(\textbf{6)}\) \(72{,}000\)
The answer is \(7.2\times10^4\).Move the decimal point \(4\) places to the left.
\(\,\,\,\,\,72{,}000\rightarrow7.2\)
Moving the decimal to the left gives a positive exponent.
\(\,\,\,\,\,72{,}000=7.2\times10^4\)
The answer is \(7.2\times10^4\).
\(\,\,\,\,\,72{,}000\rightarrow7.2\)
Moving the decimal to the left gives a positive exponent.
\(\,\,\,\,\,72{,}000=7.2\times10^4\)
The answer is \(7.2\times10^4\).
\(\textbf{7)}\) \(0.00091\)
The answer is \(9.1\times10^{-4}\).Move the decimal point \(4\) places to the right.
\(\,\,\,\,\,0.00091\rightarrow9.1\)
The original number is between \(0\) and \(1\), so the exponent is negative.
\(\,\,\,\,\,0.00091=9.1\times10^{-4}\)
The answer is \(9.1\times10^{-4}\).
\(\,\,\,\,\,0.00091\rightarrow9.1\)
The original number is between \(0\) and \(1\), so the exponent is negative.
\(\,\,\,\,\,0.00091=9.1\times10^{-4}\)
The answer is \(9.1\times10^{-4}\).
\(\textbf{8)}\) \(608{,}000{,}000\)
The answer is \(6.08\times10^8\).Move the decimal point \(8\) places to the left.
\(\,\,\,\,\,608{,}000{,}000\rightarrow6.08\)
Moving the decimal to the left gives a positive exponent.
\(\,\,\,\,\,608{,}000{,}000=6.08\times10^8\)
The answer is \(6.08\times10^8\).
\(\,\,\,\,\,608{,}000{,}000\rightarrow6.08\)
Moving the decimal to the left gives a positive exponent.
\(\,\,\,\,\,608{,}000{,}000=6.08\times10^8\)
The answer is \(6.08\times10^8\).
\(\textbf{9)}\) \(0.0562\)
The answer is \(5.62\times10^{-2}\).Move the decimal point \(2\) places to the right.
\(\,\,\,\,\,0.0562\rightarrow5.62\)
The original number is between \(0\) and \(1\), so the exponent is negative.
\(\,\,\,\,\,0.0562=5.62\times10^{-2}\)
The answer is \(5.62\times10^{-2}\).
\(\,\,\,\,\,0.0562\rightarrow5.62\)
The original number is between \(0\) and \(1\), so the exponent is negative.
\(\,\,\,\,\,0.0562=5.62\times10^{-2}\)
The answer is \(5.62\times10^{-2}\).
\(\textbf{10)}\) \(9{,}340\)
The answer is \(9.34\times10^3\).Move the decimal point \(3\) places to the left.
\(\,\,\,\,\,9{,}340\rightarrow9.34\)
Moving the decimal to the left gives a positive exponent.
\(\,\,\,\,\,9{,}340=9.34\times10^3\)
The answer is \(9.34\times10^3\).
\(\,\,\,\,\,9{,}340\rightarrow9.34\)
Moving the decimal to the left gives a positive exponent.
\(\,\,\,\,\,9{,}340=9.34\times10^3\)
The answer is \(9.34\times10^3\).
Express in decimal form
\(\textbf{11)}\) \(6.13\times10^6\)
The answer is \(6{,}130{,}000\).The exponent is positive, so move the decimal point \(6\) places to the right.
\(\,\,\,\,\,6.13\rightarrow6{,}130{,}000\)
\(\,\,\,\,\,6.13\times10^6=6{,}130{,}000\)
The answer is \(6{,}130{,}000\).
\(\,\,\,\,\,6.13\rightarrow6{,}130{,}000\)
\(\,\,\,\,\,6.13\times10^6=6{,}130{,}000\)
The answer is \(6{,}130{,}000\).
\(\textbf{12)}\) \(3.49\times10^{-5}\)
The answer is \(0.0000349\).The exponent is negative, so move the decimal point \(5\) places to the left.
\(\,\,\,\,\,3.49\rightarrow0.0000349\)
\(\,\,\,\,\,3.49\times10^{-5}=0.0000349\)
The answer is \(0.0000349\).
\(\,\,\,\,\,3.49\rightarrow0.0000349\)
\(\,\,\,\,\,3.49\times10^{-5}=0.0000349\)
The answer is \(0.0000349\).
\(\textbf{13)}\) \(7.2\times10^0\)
The answer is \(7.2\).Any nonzero number raised to the zero power equals \(1\).
\(\,\,\,\,\,10^0=1\)
\(\,\,\,\,\,7.2\times10^0=7.2\times1\)
\(\,\,\,\,\,7.2\times1=7.2\)
The answer is \(7.2\).
\(\,\,\,\,\,10^0=1\)
\(\,\,\,\,\,7.2\times10^0=7.2\times1\)
\(\,\,\,\,\,7.2\times1=7.2\)
The answer is \(7.2\).
\(\textbf{14)}\) \(4.8\times10^3\)
The answer is \(4{,}800\).The exponent is positive, so move the decimal point \(3\) places to the right.
\(\,\,\,\,\,4.8\rightarrow4{,}800\)
\(\,\,\,\,\,4.8\times10^3=4{,}800\)
The answer is \(4{,}800\).
\(\,\,\,\,\,4.8\rightarrow4{,}800\)
\(\,\,\,\,\,4.8\times10^3=4{,}800\)
The answer is \(4{,}800\).
\(\textbf{15)}\) \(2.07\times10^{-4}\)
The answer is \(0.000207\).The exponent is negative, so move the decimal point \(4\) places to the left.
\(\,\,\,\,\,2.07\rightarrow0.000207\)
\(\,\,\,\,\,2.07\times10^{-4}=0.000207\)
The answer is \(0.000207\).
\(\,\,\,\,\,2.07\rightarrow0.000207\)
\(\,\,\,\,\,2.07\times10^{-4}=0.000207\)
The answer is \(0.000207\).
\(\textbf{16)}\) \(9.5\times10^7\)
The answer is \(95{,}000{,}000\).The exponent is positive, so move the decimal point \(7\) places to the right.
\(\,\,\,\,\,9.5\rightarrow95{,}000{,}000\)
\(\,\,\,\,\,9.5\times10^7=95{,}000{,}000\)
The answer is \(95{,}000{,}000\).
\(\,\,\,\,\,9.5\rightarrow95{,}000{,}000\)
\(\,\,\,\,\,9.5\times10^7=95{,}000{,}000\)
The answer is \(95{,}000{,}000\).
\(\textbf{17)}\) \(1.26\times10^{-2}\)
The answer is \(0.0126\).The exponent is negative, so move the decimal point \(2\) places to the left.
\(\,\,\,\,\,1.26\rightarrow0.0126\)
\(\,\,\,\,\,1.26\times10^{-2}=0.0126\)
The answer is \(0.0126\).
\(\,\,\,\,\,1.26\rightarrow0.0126\)
\(\,\,\,\,\,1.26\times10^{-2}=0.0126\)
The answer is \(0.0126\).
\(\textbf{18)}\) \(3.004\times10^5\)
The answer is \(300{,}400\).The exponent is positive, so move the decimal point \(5\) places to the right.
\(\,\,\,\,\,3.004\rightarrow300{,}400\)
\(\,\,\,\,\,3.004\times10^5=300{,}400\)
The answer is \(300{,}400\).
\(\,\,\,\,\,3.004\rightarrow300{,}400\)
\(\,\,\,\,\,3.004\times10^5=300{,}400\)
The answer is \(300{,}400\).
\(\textbf{19)}\) \(8\times10^{-6}\)
The answer is \(0.000008\).The exponent is negative, so move the decimal point \(6\) places to the left.
\(\,\,\,\,\,8\rightarrow0.000008\)
\(\,\,\,\,\,8\times10^{-6}=0.000008\)
The answer is \(0.000008\).
\(\,\,\,\,\,8\rightarrow0.000008\)
\(\,\,\,\,\,8\times10^{-6}=0.000008\)
The answer is \(0.000008\).
\(\textbf{20)}\) \(5.73\times10^1\)
The answer is \(57.3\).The exponent is positive, so move the decimal point \(1\) place to the right.
\(\,\,\,\,\,5.73\rightarrow57.3\)
\(\,\,\,\,\,5.73\times10^1=57.3\)
The answer is \(57.3\).
\(\,\,\,\,\,5.73\rightarrow57.3\)
\(\,\,\,\,\,5.73\times10^1=57.3\)
The answer is \(57.3\).
Simplify, give answer in scientific notation
\(\textbf{21)}\) \((5.8\times10^4)(3.8\times10^3)\)
The answer is \(2.204\times10^8\).Multiply the coefficients and add the exponents.
\(\,\,\,\,\,(5.8)(3.8)=22.04\)
\(\,\,\,\,\,10^4\cdot10^3=10^{4+3}=10^7\)
\(\,\,\,\,\,(5.8\times10^4)(3.8\times10^3)=22.04\times10^7\)
Move the decimal point one place to the left and increase the exponent by \(1\).
\(\,\,\,\,\,22.04\times10^7=2.204\times10^8\)
The answer is \(2.204\times10^8\).
\(\,\,\,\,\,(5.8)(3.8)=22.04\)
\(\,\,\,\,\,10^4\cdot10^3=10^{4+3}=10^7\)
\(\,\,\,\,\,(5.8\times10^4)(3.8\times10^3)=22.04\times10^7\)
Move the decimal point one place to the left and increase the exponent by \(1\).
\(\,\,\,\,\,22.04\times10^7=2.204\times10^8\)
The answer is \(2.204\times10^8\).
\(\textbf{22)}\) \((5.8\times10^{-4})(3.8\times10^3)\)
The answer is \(2.204\times10^0\).Multiply the coefficients and add the exponents.
\(\,\,\,\,\,(5.8)(3.8)=22.04\)
\(\,\,\,\,\,10^{-4}\cdot10^3=10^{-4+3}=10^{-1}\)
\(\,\,\,\,\,(5.8\times10^{-4})(3.8\times10^3)=22.04\times10^{-1}\)
Move the decimal point one place to the left and increase the exponent by \(1\).
\(\,\,\,\,\,22.04\times10^{-1}=2.204\times10^0\)
The answer is \(2.204\times10^0\).
\(\,\,\,\,\,(5.8)(3.8)=22.04\)
\(\,\,\,\,\,10^{-4}\cdot10^3=10^{-4+3}=10^{-1}\)
\(\,\,\,\,\,(5.8\times10^{-4})(3.8\times10^3)=22.04\times10^{-1}\)
Move the decimal point one place to the left and increase the exponent by \(1\).
\(\,\,\,\,\,22.04\times10^{-1}=2.204\times10^0\)
The answer is \(2.204\times10^0\).
\(\textbf{23)}\) \((2.8\times10^{-4})(1.8\times10^{-3})\)
The answer is \(5.04\times10^{-7}\).Multiply the coefficients and add the exponents.
\(\,\,\,\,\,(2.8)(1.8)=5.04\)
\(\,\,\,\,\,10^{-4}\cdot10^{-3}=10^{-4+(-3)}\)
\(\,\,\,\,\,10^{-4+(-3)}=10^{-7}\)
\(\,\,\,\,\,(2.8\times10^{-4})(1.8\times10^{-3})=5.04\times10^{-7}\)
The answer is \(5.04\times10^{-7}\).
\(\,\,\,\,\,(2.8)(1.8)=5.04\)
\(\,\,\,\,\,10^{-4}\cdot10^{-3}=10^{-4+(-3)}\)
\(\,\,\,\,\,10^{-4+(-3)}=10^{-7}\)
\(\,\,\,\,\,(2.8\times10^{-4})(1.8\times10^{-3})=5.04\times10^{-7}\)
The answer is \(5.04\times10^{-7}\).
\(\textbf{24)}\) \(\displaystyle\frac{1.6\times10^5}{3.2\times10^3}\)
The answer is \(5\times10^1\).Divide the coefficients and subtract the exponents.
\(\,\,\,\,\,\frac{1.6}{3.2}=0.5\)
\(\,\,\,\,\,\frac{10^5}{10^3}=10^{5-3}=10^2\)
\(\,\,\,\,\,\frac{1.6\times10^5}{3.2\times10^3}=0.5\times10^2\)
Move the decimal point one place to the right and decrease the exponent by \(1\).
\(\,\,\,\,\,0.5\times10^2=5\times10^1\)
The answer is \(5\times10^1\).
\(\,\,\,\,\,\frac{1.6}{3.2}=0.5\)
\(\,\,\,\,\,\frac{10^5}{10^3}=10^{5-3}=10^2\)
\(\,\,\,\,\,\frac{1.6\times10^5}{3.2\times10^3}=0.5\times10^2\)
Move the decimal point one place to the right and decrease the exponent by \(1\).
\(\,\,\,\,\,0.5\times10^2=5\times10^1\)
The answer is \(5\times10^1\).
\(\textbf{25)}\) \((4.6\times10^5)+(3.2\times10^5)\)
The answer is \(7.8\times10^5\).The powers of \(10\) are already the same, so add the coefficients.
\(\,\,\,\,\,(4.6+3.2)\times10^5\)
\(\,\,\,\,\,7.8\times10^5\)
The coefficient is between \(1\) and \(10\), so the answer is in scientific notation.
The answer is \(7.8\times10^5\).
\(\,\,\,\,\,(4.6+3.2)\times10^5\)
\(\,\,\,\,\,7.8\times10^5\)
The coefficient is between \(1\) and \(10\), so the answer is in scientific notation.
The answer is \(7.8\times10^5\).
\(\textbf{26)}\) \((5.4\times10^{-3})-(2.8\times10^{-3})\)
The answer is \(2.6\times10^{-3}\).The powers of \(10\) are already the same, so subtract the coefficients.
\(\,\,\,\,\,(5.4-2.8)\times10^{-3}\)
\(\,\,\,\,\,2.6\times10^{-3}\)
The coefficient is between \(1\) and \(10\), so the answer is in scientific notation.
The answer is \(2.6\times10^{-3}\).
\(\,\,\,\,\,(5.4-2.8)\times10^{-3}\)
\(\,\,\,\,\,2.6\times10^{-3}\)
The coefficient is between \(1\) and \(10\), so the answer is in scientific notation.
The answer is \(2.6\times10^{-3}\).
\(\textbf{27)}\) \((1.6\times10^5)+(3.2\times10^3)\)
The answer is \(1.632\times10^5\).The powers of \(10\) must be the same before adding.
Rewrite \(3.2\times10^3\) using \(10^5\).
\(\,\,\,\,\,3.2\times10^3=0.032\times10^5\)
\(\,\,\,\,\,(1.6\times10^5)+(0.032\times10^5)\)
\(\,\,\,\,\,(1.6+0.032)\times10^5\)
\(\,\,\,\,\,1.632\times10^5\)
The answer is \(1.632\times10^5\).
Rewrite \(3.2\times10^3\) using \(10^5\).
\(\,\,\,\,\,3.2\times10^3=0.032\times10^5\)
\(\,\,\,\,\,(1.6\times10^5)+(0.032\times10^5)\)
\(\,\,\,\,\,(1.6+0.032)\times10^5\)
\(\,\,\,\,\,1.632\times10^5\)
The answer is \(1.632\times10^5\).
\(\textbf{28)}\) \((1.6\times10^5)-(3.2\times10^4)\)
The answer is \(1.28\times10^5\).The powers of \(10\) must be the same before subtracting.
Rewrite \(3.2\times10^4\) using \(10^5\).
\(\,\,\,\,\,3.2\times10^4=0.32\times10^5\)
\(\,\,\,\,\,(1.6\times10^5)-(0.32\times10^5)\)
\(\,\,\,\,\,(1.6-0.32)\times10^5\)
\(\,\,\,\,\,1.28\times10^5\)
The answer is \(1.28\times10^5\).
Rewrite \(3.2\times10^4\) using \(10^5\).
\(\,\,\,\,\,3.2\times10^4=0.32\times10^5\)
\(\,\,\,\,\,(1.6\times10^5)-(0.32\times10^5)\)
\(\,\,\,\,\,(1.6-0.32)\times10^5\)
\(\,\,\,\,\,1.28\times10^5\)
The answer is \(1.28\times10^5\).
\(\textbf{29)}\) \((5\times10^5)+(3\times10^4)\)
The answer is \(5.3\times10^5\).The powers of \(10\) must be the same before adding.
Rewrite \(3\times10^4\) using \(10^5\).
\(\,\,\,\,\,3\times10^4=0.3\times10^5\)
\(\,\,\,\,\,(5\times10^5)+(0.3\times10^5)\)
\(\,\,\,\,\,(5+0.3)\times10^5\)
\(\,\,\,\,\,5.3\times10^5\)
The answer is \(5.3\times10^5\).
Rewrite \(3\times10^4\) using \(10^5\).
\(\,\,\,\,\,3\times10^4=0.3\times10^5\)
\(\,\,\,\,\,(5\times10^5)+(0.3\times10^5)\)
\(\,\,\,\,\,(5+0.3)\times10^5\)
\(\,\,\,\,\,5.3\times10^5\)
The answer is \(5.3\times10^5\).
\(\textbf{30)}\) \((6\times10^7)-(4\times10^5)\)
The answer is \(5.96\times10^7\).The powers of \(10\) must be the same before subtracting.
Rewrite \(4\times10^5\) using \(10^7\).
\(\,\,\,\,\,4\times10^5=0.04\times10^7\)
\(\,\,\,\,\,(6\times10^7)-(0.04\times10^7)\)
\(\,\,\,\,\,(6-0.04)\times10^7\)
\(\,\,\,\,\,5.96\times10^7\)
The answer is \(5.96\times10^7\).
Rewrite \(4\times10^5\) using \(10^7\).
\(\,\,\,\,\,4\times10^5=0.04\times10^7\)
\(\,\,\,\,\,(6\times10^7)-(0.04\times10^7)\)
\(\,\,\,\,\,(6-0.04)\times10^7\)
\(\,\,\,\,\,5.96\times10^7\)
The answer is \(5.96\times10^7\).
\(\textbf{31)}\) \(\displaystyle\frac{3.6\times10^8}{1.2\times10^{-2}}\)
The answer is \(3\times10^{10}\).Divide the coefficients and subtract the exponents.
\(\,\,\,\,\,\frac{3.6}{1.2}=3\)
\(\,\,\,\,\,\frac{10^8}{10^{-2}}=10^{8-(-2)}\)
\(\,\,\,\,\,10^{8-(-2)}=10^{10}\)
\(\,\,\,\,\,\frac{3.6\times10^8}{1.2\times10^{-2}}=3\times10^{10}\)
The answer is \(3\times10^{10}\).
\(\,\,\,\,\,\frac{3.6}{1.2}=3\)
\(\,\,\,\,\,\frac{10^8}{10^{-2}}=10^{8-(-2)}\)
\(\,\,\,\,\,10^{8-(-2)}=10^{10}\)
\(\,\,\,\,\,\frac{3.6\times10^8}{1.2\times10^{-2}}=3\times10^{10}\)
The answer is \(3\times10^{10}\).
\(\textbf{32)}\) \((2.5\times10^{-4})^2\)
The answer is \(6.25\times10^{-8}\).Square the coefficient and multiply the exponents.
\(\,\,\,\,\,(2.5)^2=6.25\)
\(\,\,\,\,\,(10^{-4})^2=10^{(-4)(2)}\)
\(\,\,\,\,\,10^{(-4)(2)}=10^{-8}\)
\(\,\,\,\,\,(2.5\times10^{-4})^2=6.25\times10^{-8}\)
The answer is \(6.25\times10^{-8}\).
\(\,\,\,\,\,(2.5)^2=6.25\)
\(\,\,\,\,\,(10^{-4})^2=10^{(-4)(2)}\)
\(\,\,\,\,\,10^{(-4)(2)}=10^{-8}\)
\(\,\,\,\,\,(2.5\times10^{-4})^2=6.25\times10^{-8}\)
The answer is \(6.25\times10^{-8}\).