Perform Operations With Scientific Notation
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8th Grade Math › Perform Operations With Scientific Notation
A microscope slide has $6\times10^8$ cells spread evenly across $2\times10^5$ tiny squares on a grid. About how many cells are in each square? Write your answer in scientific notation.
$3\times10^4$
$12\times10^3$
$3\times10^3$
$3\times10^{13}$
Explanation
This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add). For this problem, divide $6×10^8$ by $2×10^5$, so (6 ÷ 2) × 10^(8-5) = 3 × $10^3$. This correct application estimates cells per square in scientific notation as $3×10^3$. A common error is adding exponents instead of subtracting for division. Steps: (1) identify operation as division, (2) for ×/÷: coefficients and exponents separately, (3) divide coefficients and subtract exponents, (4) verify proper form (1≤a<10, here it's fine), (5) no units specified. Common errors: mixing operation rules (adding exponents for addition), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.
A student measures two lengths of wire: $0.0045$ meters and $3.2\times10^{-3}$ meters. What is the total length in scientific notation?
$7.7\times10^{-3}$
$7.7\times10^{3}$
$0.77\times10^{-2}$
$4.82\times10^{-3}$
Explanation
This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add). For this problem, first convert 0.0045 to $4.5×10^{-3}$, then add to $3.2×10^{-3}$, so (4.5 + 3.2) × $10^{-3}$ = 7.7 × $10^{-3}$. This correct application gives the total length in scientific notation as $7.7×10^{-3}$ meters. A common error is adding without converting to scientific notation first. Steps: (1) identify operation as addition, (2) for +/-: adjust to same exponent first (already same), (3) add coefficients, (4) verify proper form (1≤a<10, adjust if needed), (5) include units like meters. Common errors: mixing operation rules (adding exponents for addition), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.
A sports field is $1.2\times10^2$ meters long. A student walks the length of the field $3.5\times10^1$ times. How many meters does the student walk in total? Write your answer in scientific notation.
$4.2\times10^3$
$4.2\times10^1$
$42\times10^2$
$4.7\times10^3$
Explanation
This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add). For this problem, multiply $1.2×10^2$ by $3.5×10^1$, so (1.2 × 3.5) × 10^(2+1) = 4.2 × $10^3$. This correct application gives the total distance in scientific notation as $4.2×10^3$ meters. A common error is subtracting exponents instead of adding for multiplication. Steps: (1) identify operation as multiplication, (2) for ×/÷: coefficients and exponents separately, (3) multiply coefficients and add exponents, (4) verify proper form (1≤a<10, here it's fine), (5) include units like meters. Common errors: mixing operation rules (adding exponents for addition), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.
A planet model in a classroom is scaled so that $1.5\times10^3$ km in real life becomes $3\times10^{-2}$ meters in the model. What is the product $\left(1.5\times10^3\right)\left(3\times10^{-2}\right)$ written in scientific notation?
$4.5\times10^{1}$
$1.8\times10^{1}$
$45\times10^{0}$
$4.5\times10^{-5}$
Explanation
This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add). For this problem, multiply $1.5×10^3$ by $3×10^{-2}$, so (1.5 × 3) × 10^(3-2) = 4.5 × $10^1$. This correct application gives the product in scientific notation as $4.5×10^1$. A common error is adding exponents instead of subtracting the negative one. Steps: (1) identify operation as multiplication, (2) for ×/÷: coefficients and exponents separately, (3) multiply coefficients and add exponents (including negatives), (4) verify proper form (1≤a<10, here it's fine), (5) no units here. Common errors: mixing operation rules (adding exponents for addition), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.
In a science lab, a student counts about $6\times10^6$ bacteria in one sample and $4\times10^5$ bacteria in another sample. About how many bacteria are there in total? Write your answer in scientific notation.
$1.0\times10^{12}$
$6.04\times10^6$
$6.4\times10^6$
$10\times10^6$
Explanation
This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add). For this problem, add $6×10^6$ and $4×10^5$ by adjusting the smaller to $0.4×10^6$, so (6 + 0.4) × $10^6$ = 6.4 × $10^6$. This correct application estimates the total bacteria in scientific notation as $6.4×10^6$. A common error is not adjusting exponents before adding, like just adding coefficients and exponents separately. Steps: (1) identify operation as addition, (2) for +/-: adjust to same exponent first (shift decimal), (3) add coefficients, (4) verify proper form (1≤a<10, adjust if needed: 12×10⁵→1.2×10⁶), (5) no units here. Common errors: mixing operation rules (adding exponents for addition), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.
A computer file is $7.2\times10^6$ bytes. A smaller file is $3.5\times10^5$ bytes. What is the difference in size, in scientific notation? $$\left(7.2\times10^6\right)-\left(3.5\times10^5\right)$$
$3.7\times10^1$
$7.55\times10^6$
$6.85\times10^1$
$6.85\times10^6$
Explanation
This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add). For this problem, subtract $3.5×10^5$ from $7.2×10^6$ by adjusting to $0.35×10^6$, so (7.2 - 0.35) × $10^6$ = 6.85 × $10^6$. This correct application gives the difference in scientific notation as $6.85×10^6$ bytes. A common error is subtracting exponents instead of adjusting to the same power. Steps: (1) identify operation as subtraction, (2) for +/-: adjust to same exponent first (shift decimal), (3) subtract coefficients, (4) verify proper form (1≤a<10, here it's fine), (5) include units like bytes. Common errors: mixing operation rules (adding exponents for addition), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.
A school is tracking recycling. On Monday, students collected $3.2\times10^5$ grams of paper, and on Tuesday they collected $4.8\times10^5$ grams. What is the total amount collected, in scientific notation?
$0.8\times10^6$
$8\times10^{10}$
$8\times10^0$
$8.0\times10^5$
Explanation
This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add). For this problem, add $3.2×10^5$ and $4.8×10^5$, which already have the same exponent, so (3.2 + 4.8) × $10^5$ = 8.0 × $10^5$. This correct application gives the total in scientific notation as $8.0×10^5$. A common error is treating addition like multiplication by adding exponents instead of adjusting to the same power. Steps: (1) identify operation as addition, (2) for +/-: adjust to same exponent first (not needed here), (3) add coefficients, (4) verify proper form (1≤a<10, here it's fine), (5) include units if given, like grams. Common errors: mixing operation rules (adding exponents for addition), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.
A student simplifies the expression for a science project: $$\frac{\left(4.8\times10^7\right)\left(2.5\times10^{-3}\right)}{6\times10^2}.$$ What is the value in scientific notation?
$2.0\times10^{-2}$
$2.0\times10^2$
$20\times10^1$
$2.0\times10^{0}$
Explanation
This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: $(2\times10^3)\times(4\times10^5)=(2\times4)\times10^{3+5}=8\times10^8$ (multiply coefficients, add exponents). Division: $(6\times10^8)\div(2\times10^5)=(6\div2)\times10^{8-5}=3\times10^3$ (divide coefficients, subtract exponents). Addition: requires same exponent—$(3\times10^5)+(2\times10^4)=3\times10^5+0.2\times10^5=3.2\times10^5$ (convert $2\times10^4$ to $0.2\times10^5$, then add). For this problem, compute numerator $(4.8\times10^7)(2.5\times10^{-3}) = 1.2\times10^5$, then divide by $6\times10^2$ to get $2.0\times10^2$. This correct application simplifies the expression in scientific notation as $2.0\times10^2$. A common error is mishandling the exponents in combined operations. Steps: (1) identify operations as multiplication then division, (2) for $\times$/÷: coefficients and exponents separately, (3) handle numerator first, then divide, (4) verify proper form ($1\leq a<10$, adjust if needed), (5) no units here. Common errors: mixing operation rules (adding exponents for addition), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.
A science class measured two distances for a model rocket launch: $6.0\times10^6$ cm and $4.0\times10^5$ cm. Find the total distance, written in scientific notation.
$6.4\times10^6$
$6.04\times10^6$
$1.0\times10^{12}$
$10.0\times10^6$
Explanation
This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add). For this problem, add $6.0×10^6$ and $4.0×10^5$; convert $4.0×10^5$ to $0.4×10^6$, then sum: 6.0 + 0.4 = $6.4×10^6$ cm. This correct application involves adjusting the smaller exponent by moving the decimal left and increasing the exponent. A common error is adding without adjustment, like 6.0 + 4.0 = $10.0×10^{something}$, but that ignores exponents. Steps: (1) identify operation as addition, (2) for ×/÷: not applicable, (3) for +/-: adjust to same exponent first (shift decimal), (4) verify proper form (1≤a<10, 6.4 is fine), (5) include units if given, here cm but not in final answer. Common errors: mixing operation rules (adding exponents for addition), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.
A microscope slide has $2.5\times10^3$ bacteria in one drop and $4.0\times10^5$ bacteria in another drop. If the drops are combined, how many bacteria are there in total? Write your answer in scientific notation.
$4.25\times10^5$
$4.00025\times10^5$
$4.025\times10^5$
$6.5\times10^8$
Explanation
This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add). For this problem, add $2.5×10^3$ and $4.0×10^5$; convert $2.5×10^3$ to $0.025×10^5$, then sum: 4.0 + 0.025 = $4.025×10^5$ bacteria. This correct application adjusts the smaller number by decreasing the coefficient and increasing the exponent by 2. A common error is not adjusting and just adding coefficients with the larger exponent, like 4.0 + 2.5 = $6.5×10^5$, ignoring the difference in magnitude. Steps: (1) identify operation as addition, (2) for ×/÷: not applicable, (3) for +/-: adjust to same exponent first (shift decimal), (4) verify proper form (1≤a<10, 4.025 is fine), (5) no units beyond bacteria. Common errors: mixing operation rules (adding exponents for addition), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.