Question

addition or subtraction
(2.3×10^6)+(1.6×10^9)
(2.3×10^6)+(1.6×10^3)×10^6
(2.3×10^6)+(1,600×10^6)
(2.3 + 1,600)×10^6
1,602.3×10^6
1.6023×10^9
use the product of powers law.
multiplication
(2.3×10^6)×(1.6×10^9)
(2.3×1.6)×(10^6×10^9)
3.68×10^(6 + 9)
3.68×10^15
use the product of powers law.
(2.3×10^6)÷(1.6×10^9)
(2.3÷1.6)×(10^6÷10^9)
1.4375×10^(6 - 9)
1.4375×10^(-3)
use the quotient of powers law.
when physical measurements are involved, the answer should have the same number of significant digits as the given measurement with the least number of significant digits.
do you understand?

1. ? essential question how does using scientific notation help when computing with very small or very large numbers?
2. check for reasonableness when working with physical measurements why is the number of significant digits in the answer important?
3. use patterns and structure for the sum of (5.2×10^4) and (6.95×10^4) in scientific notation, why will the power of 10 be 10^5?

do you know how?

1. a bacteriologist estimates that there are 5.2×10^4 bacteria growing in each of 20 petri dishes. about how many bacteria in total are growing in the petri dishes? express your answer in scientific notation.
2. the distance from earth to the moon is approximately 1.2×10^9 feet. the apollo 11 spacecraft was approximately 360 feet long. about how many spacecraft of that length would fit end to end from earth to the moon? express your answer in scientific notation.
3. the mass of mars is 6.42×10^23 kilograms. the mass of mercury is 3.3×10^23 kilograms.

a. what is the combined mass of mars and mercury expressed in scientific notation?
b. how many significant digits should be in the solution?

Explanation:

Paso 1: Resolver la pregunta 4

Para encontrar el número total de bacterias, multiplicamos el número de bacterias en cada plato de培养皿 ($5.2\times10^{4}$) por el número de platos de培养皿 (20).
$20 = 2\times10^{1}$
$(5.2\times10^{4})\times(2\times10^{1})=(5.2\times2)\times(10^{4}\times10^{1})$
$10.4\times10^{4 + 1}=10.4\times10^{5}$
Convertimos a notación científica correcta: $1.04\times10^{6}$

Paso 2: Resolver la pregunta 5

Para encontrar el número de naves espaciales que caben, dividimos la distancia de la Tierra a la Luna ($1.2\times10^{9}$ pies) entre la longitud de la nave espacial (360 pies = $3.6\times10^{2}$ pies).
$(1.2\times10^{9})\div(3.6\times10^{2})=(1.2\div3.6)\times(10^{9}\div10^{2})$
$\frac{1.2}{3.6}\times10^{9 - 2}=\frac{1}{3}\times10^{7}\approx0.333\times10^{7}$
Convertimos a notación científica correcta: $3.33\times10^{6}$

Paso 3: Resolver la pregunta 6a

Para encontrar la masa combinada de Marte y Mercurio, sumamos sus masas.
$(6.42\times10^{23})+(3.3\times10^{23})=(6.42 + 3.3)\times10^{23}=9.72\times10^{23}$

Paso 4: Resolver la pregunta 6b

El número con menos cifras significativas entre $6.42\times10^{23}$ y $3.3\times10^{23}$ es $3.3\times10^{23}$ que tiene 2 cifras significativas. Entonces, la solución debe tener 2 cifras significativas.

Respuesta:

1. $1.04\times10^{6}$
2. $3.33\times10^{6}$

6a. $9.72\times10^{23}$
6b. 2

Answer:

Paso 1: Resolver la pregunta 4

Para encontrar el número total de bacterias, multiplicamos el número de bacterias en cada plato de培养皿 ($5.2\times10^{4}$) por el número de platos de培养皿 (20).
$20 = 2\times10^{1}$
$(5.2\times10^{4})\times(2\times10^{1})=(5.2\times2)\times(10^{4}\times10^{1})$
$10.4\times10^{4 + 1}=10.4\times10^{5}$
Convertimos a notación científica correcta: $1.04\times10^{6}$

Paso 2: Resolver la pregunta 5

Para encontrar el número de naves espaciales que caben, dividimos la distancia de la Tierra a la Luna ($1.2\times10^{9}$ pies) entre la longitud de la nave espacial (360 pies = $3.6\times10^{2}$ pies).
$(1.2\times10^{9})\div(3.6\times10^{2})=(1.2\div3.6)\times(10^{9}\div10^{2})$
$\frac{1.2}{3.6}\times10^{9 - 2}=\frac{1}{3}\times10^{7}\approx0.333\times10^{7}$
Convertimos a notación científica correcta: $3.33\times10^{6}$

Paso 3: Resolver la pregunta 6a

Para encontrar la masa combinada de Marte y Mercurio, sumamos sus masas.
$(6.42\times10^{23})+(3.3\times10^{23})=(6.42 + 3.3)\times10^{23}=9.72\times10^{23}$

Paso 4: Resolver la pregunta 6b

El número con menos cifras significativas entre $6.42\times10^{23}$ y $3.3\times10^{23}$ es $3.3\times10^{23}$ que tiene 2 cifras significativas. Entonces, la solución debe tener 2 cifras significativas.

Respuesta:

1. $1.04\times10^{6}$
2. $3.33\times10^{6}$

6a. $9.72\times10^{23}$
6b. 2