Free math worksheets

- Express the scale factor as a power of \(10\). - Keep the normalized coefficient unchanged. - Adjust the exponent by the number of place-value steps.

1. Since \(100=10^2\), multiply by \(10^2\). 2. \(3.7\times10^{-8}\cdot10^2=3.7\times10^{-6}\). 3. The coefficient stays \(3.7\), and the exponent increases by \(2\).

\(3.7\times10^{-6}\)

- Translate “one thousandth” into a power of \(10\). - Adjust the existing exponent. - Expand the normalized result.

1. One thousandth is \(10^{-3}\). 2. \(8.2\times10^5\cdot10^{-3}=8.2\times10^2\). 3. In standard form, \(8.2\times10^2=820\).

\(8.2\times10^2=820\)

- Apply the exponent to both factors inside the parentheses. - Use the power-of-a-power rule for the power of \(10\). - Check whether the resulting coefficient is normalized.

1. Square the coefficient: \(3^2=9\). 2. Apply the power-of-a-power rule: \((10^4)^2=10^8\). 3. Therefore, \((3\times10^4)^2=9\times10^8\), which is normalized.

\(9\times10^8\)

- Divide coefficients and powers of \(10\) separately. - Use the exponent rule for division with equal bases. - Confirm that the quotient coefficient is in the required range.

1. Divide the coefficients: \(8.4\div2.1=4\). 2. Divide the powers of \(10\): \(10^7\div10^3=10^4\). 3. Therefore, the quotient is \(4\times10^4\).

\(4\times10^4\)

- Check whether the place-value factors match. - Combine only the coefficients when the powers are equal. - Keep the common power of \(10\).

1. The powers of \(10\) match, so add the coefficients. 2. \(4.7+2.8=7.5\). 3. Therefore, the sum is \(7.5\times10^6\).

\(7.5\times10^6\)

- Verify that both quantities use the same exponent. - Subtract the coefficients while preserving the common power. - Check the sign and normalization of the result.

1. The powers of \(10\) match, so subtract the coefficients. 2. \(9.1-3.6=5.5\). 3. Therefore, the difference is \(5.5\times10^{-5}\).

\(5.5\times10^{-5}\)

- Equal repeated quantities can be combined with multiplication. - The whole-number factor does not introduce another power of \(10\). - Keep the area unit unchanged.

1. Multiply the area of one panel by \(6\): \(6(1.25\times10^{-2})\,\text{m}^2\). 2. \(6\cdot1.25=7.5\), so the combined area is \(7.5\times10^{-2}\,\text{m}^2\).

\(7.5\times10^{-2}\,\text{m}^2\)

- Translate each calculator display into coefficient-and-power form. - Combine the two factors before writing standard notation. - Check whether the resulting coefficient is within the required range.

1. Interpret the display as \((2.5\times10^6)(4\times10^{-3})\). 2. Multiply: \(10\times10^3\). 3. Normalize: \(10\times10^3=1\times10^4\).

\(1\times10^4\)

- Identify whether each power of \(10\) is being multiplied or is part of an addend. - For addition, ask whether the coefficients currently refer to the same place-value unit. - Relate the multiplication case to equal-base exponent laws and the addition case to like place values.

1. a) In A, the powers are factors, so \(10^4\cdot10^{-3}=10^{4+(-3)}=10^1\). 2. b) One valid rewrite is \(2\times10^3=0.2\times10^4\), so both addends use \(10^4\). Equivalently, \(3\times10^4=30\times10^3\), so both addends use \(10^3\). 3. c) Multiplication combines powers as factors, so equal-base exponent laws apply. Addition combines quantities by place value, so the powers of \(10\) must represent the same place-value unit before the coefficients can be added.

a) Add the exponents: \(4+(-3)=1\). b) For example, \(2\times10^3=0.2\times10^4\). Equivalently, \(3\times10^4=30\times10^3\). c) Multiplication combines power factors, while addition requires matching place-value units first.

- The needed unit-conversion factors are supplied, so focus on whether each conversion multiplies or divides by \(10^3\). - Apply the exponent rule for multiplying or dividing powers of \(10\). - Check that every final coefficient is at least \(1\) and less than \(10\).

1. a) To convert to cubic millimeters, multiply by \(10^3\): \((2.4\times10^3)\cdot10^3=2.4\times10^6\,\text{mm}^3\). To convert to cubic decimeters, divide by \(10^3\): \(\frac{2.4\times10^3}{10^3}=2.4\times10^0\,\text{dm}^3\). 2. b) To convert to cubic centimeters, multiply by \(10^3\): \((5.0\times10^{-2})\cdot10^3=5.0\times10^1\,\text{cm}^3\). To convert to cubic meters, divide by \(10^3\): \(\frac{5.0\times10^{-2}}{10^3}=5.0\times10^{-5}\,\text{m}^3\). 3. c) Since \(1\,\text{L}=1\,\text{dm}^3\), start with \(7.1\times10^0\,\text{dm}^3\). Multiplying by \(10^3\) gives \(7.1\times10^3\,\text{cm}^3\), and dividing by \(10^3\) gives \(7.1\times10^{-3}\,\text{m}^3\).

a) \((2.4\times10^3)\cdot10^3=2.4\times10^6\,\text{mm}^3\); \(\frac{2.4\times10^3}{10^3}=2.4\times10^0\,\text{dm}^3\) b) \((5.0\times10^{-2})\cdot10^3=5.0\times10^1\,\text{cm}^3\); \(\frac{5.0\times10^{-2}}{10^3}=5.0\times10^{-5}\,\text{m}^3\) c) \((7.1\times10^0)\cdot10^3=7.1\times10^3\,\text{cm}^3\); \(\frac{7.1\times10^0}{10^3}=7.1\times10^{-3}\,\text{m}^3\)

- Convert both measurements to meters first. - Divide the larger thickness by the smaller thickness. - Subtract exponents when dividing powers of \(10\).

1. Convert the coating thickness: \(150\,\text{nm}=150\times10^{-9}\,\text{m}=1.5\times10^{-7}\,\text{m}\). 2. Convert the hair thickness: \(0.06\,\text{mm}=0.06\times10^{-3}\,\text{m}=6\times10^{-5}\,\text{m}\). 3. Divide the thicknesses: \(\frac{6\times10^{-5}}{1.5\times10^{-7}}=\frac{6}{1.5}\times10^{-5-(-7)}=4\times10^2=400\).

Coating: \(1.5\times10^{-7}\,\text{m}\). Hair: \(6\times10^{-5}\,\text{m}\). Approximately \(400\) layers.

- Convert both distances to kilometers. - Multiply the light-year conversion by \(12\). - Compare the coefficients when the powers of ten are the same.

1. Using the given approximation, estimate Star A's distance: \(12\cdot(9.5\times 10^{12})=114\times 10^{12}=1.14\times 10^{14}\,\text{km}\). Thus, Star A is approximately \(1.14\times 10^{14}\,\text{km}\) away. 2. Compare the distances: \(1.14\times 10^{14}>1.0\times 10^{14}\). 3. Therefore, Star A is farther from Earth.

Star A is farther away because its distance is approximately \(1.14\times 10^{14}\,\text{km}\), which is greater than \(1.0\times 10^{14}\,\text{km}\).

- Form a quotient of the larger estimate and the smaller estimate. - Compare coefficients and powers of \(10\) separately. - Express the result as an ordinary multiplicative comparison.

1. Compare the coefficients: \(4\div2=2\). 2. Compare the powers of \(10\): \(10^9\div10^6=10^3\). 3. The ratio is \(2\times10^3=2000\). 4. The archive holds about \(2000\) times as many bytes.

About \(2000\) times as many

- Divide the larger quantity by the smaller one. - Treat coefficients and powers of \(10\) separately. - Convert the final scientific-notation ratio to a whole number.

1. Form the quotient \(\frac{3\times10^{-4}}{6\times10^{-7}}\). 2. The coefficient quotient is \(\frac{3}{6}=\frac{1}{2}\). 3. The power quotient is \(10^{-4-(-7)}=10^3\). 4. Thus, the ratio is \(\frac{1}{2}\times10^3=500\).

\(500\) times as large

- Multiply coefficients and powers separately. - Combine the exponents before normalizing. - Adjust a coefficient greater than or equal to \(10\).

1. Multiply coefficients: \(6\cdot4=24\). 2. Add exponents: \(10^7\cdot10^{-3}=10^4\). 3. The product is \(24\times10^4\). 4. Normalize: \(24\times10^4=2.4\times10^5\).

\(2.4\times10^5\)

- Divide the coefficient and power factors separately. - Be careful when subtracting a positive exponent from a negative exponent. - Renormalize a coefficient below \(1\).

1. Divide coefficients: \(3.6\div9=0.4\). 2. Subtract exponents: \(10^{-8}\div10^2=10^{-10}\). 3. The quotient is \(0.4\times10^{-10}\). 4. Normalize: \(0.4\times10^{-10}=4\times10^{-11}\).

\(4\times10^{-11}\)

- Rewrite one addend so both powers of \(10\) match. - Preserve place value while changing the coefficient. - Add only after the exponents agree.

1. Rewrite \(7.5\times10^4=0.75\times10^5\). 2. Add coefficients: \(3.2+0.75=3.95\). 3. The sum is \(3.95\times10^5\).

\(3.95\times10^5\)

- Express both terms using one common power of \(10\). - Check which term is larger before subtracting. - Normalize the result if needed.

1. Rewrite \(8\times10^{-4}=0.8\times10^{-3}\). 2. Subtract coefficients: \(6-0.8=5.2\). 3. The difference is \(5.2\times10^{-3}\).

\(5.2\times10^{-3}\)

- Convert the decimal addend to scientific notation first. - Use the same exponent as the existing term. - Add the coefficients once the forms match.

1. Convert \(0.00045=4.5\times10^{-4}\). 2. Add: \(4.5\times10^{-4}+2.7\times10^{-4}=7.2\times10^{-4}\).

\(7.2\times10^{-4}\)

- Represent both quantities in compatible forms. - Separate the numerical factors from the powers of \(10\). - Check that the result is smaller than an hour's output.

1. Convert \(0.003=3\times10^{-3}\). 2. Multiply: \((2.4\times10^6)(3\times10^{-3})=7.2\times10^3\).

\(7.2\times10^3\) records

- Put the total and the portion size in the same notation. - Think about how many small portions fit in the total. - Verify that the count is a whole number.

1. Convert \(0.00084=8.4\times10^{-4}\). 2. Divide: \(\frac{8.4\times10^{-4}}{2.1\times10^{-6}}=4\times10^2\).

\(4\times10^2\) portions

- Treat multiplication, division, and addition as three different structural cases rather than trying to use one exponent rule for all of them. - For the quotient, pay close attention to subtracting a negative exponent. - For the sum, the coefficients can be combined only after both addends use the same power of \(10\).

1. A. For multiplication, add the exponents of the powers of \(10\): \((1.2\cdot3)\times10^{5+(-2)}=3.6\times10^3\). 2. B. For division, subtract the exponent in the denominator from the exponent in the numerator: \(\frac{8.4}{2.1}\times10^{-6-(-3)}=4\times10^{-3}\). 3. C. For addition, first express the addends with the same power of \(10\): \(7.5\times10^3=0.75\times10^4\). Then \((6.5+0.75)\times10^4=7.25\times10^4\).

A. Add the exponents; \(3.6\times10^3\) B. Subtract the denominator exponent from the numerator exponent; \(4\times10^{-3}\) C. Align the powers of \(10\) first; \(7.25\times10^4\)

- Express both counts with a common power of \(10\). - Expect a result much smaller than either starting count. - Normalize only after finding the difference.

1. Rewrite \(9.8\times10^4=0.98\times10^5\). 2. Subtract: \((1.02-0.98)\times10^5=0.04\times10^5\). 3. Normalize: \(0.04\times10^5=4\times10^3\).

\(4\times10^3\) items

- Separate the coefficient multiplication from the power-of-ten multiplication. - Pay attention to the sign on the second exponent. - Check the allowed size of a normalized coefficient.

1. When multiplying powers of \(10\), add the exponents: \(3+(-2)=1\), not \(5\). 2. The unnormalized product is \(20\times10^1\). 3. Since a scientific-notation coefficient must be less than \(10\), normalize to \(2\times10^2\).

The exponent should first be \(1\), and \(20\times10^1\) is not normalized. The correct product is \(2\times10^2\).

- Verify the lowest operation before checking the top operation. - Compare the powers in the quotient to judge the expected magnitude. - Distinguish a calculation error from a notation-format issue.

1. The lower product is \((6\times10^4)(3\times10^{-2})=18\times10^2=1.8\times10^3\), so its displayed result is correct. 2. The final quotient is \(\frac{1.8\times10^3}{9\times10^1}=0.2\times10^2=2\times10^1\). 3. Therefore the displayed final result \(2\times10^{-1}\) is incorrect.

The incorrect result is the final \(2\text{E}{-1}\). It should be \(2\times10^1\).

- Choose one power-of-ten unit for all three quantities. - Track the received and shipped amounts with opposite operations. - Estimate the result before combining exact coefficients.

1. Rewrite all terms with \(10^6\): \(4\times10^5=0.4\times10^6\) and \(7\times10^4=0.07\times10^6\). 2. Combine coefficients: \(2.5+0.4-0.07=2.83\). 3. The remaining count is \(2.83\times10^6\).

\(2.83\times10^6\) items

- Undo the multiplication to isolate the unknown factor. - Treat the coefficient and power-of-ten parts separately. - Check the result by multiplying it back into the equation.

1. Divide the product by the known factor: \(x=\frac{1.2\times10^9}{3\times10^5}\). 2. Divide coefficients and powers: \(x=0.4\times10^4\). 3. Normalize: \(x=4\times10^3\).

\(x=4\times10^3\)

- Use the relationship among a dividend, divisor, and quotient. - Decide whether the unknown should be larger or smaller than the numerator. - Substitute your result to confirm the quotient.

1. Rearrange the equation: \(x=\frac{7.2\times10^{-4}}{9\times10^{-2}}\). 2. Divide: \(x=0.8\times10^{-2}\). 3. Normalize: \(x=8\times10^{-3}\).

\(x=8\times10^{-3}\)

- Evaluate each product before comparing them. - Normalize both results to make their magnitudes easy to compare. - Express both values with a common exponent before finding the difference.

1. Sensor A gives \((7.5\times10^3)(4\times10^{-5})=30\times10^{-2}=3\times10^{-1}\). 2. Sensor B gives \((2.4\times10^2)(9\times10^{-4})=21.6\times10^{-2}=2.16\times10^{-1}\). 3. Subtract: \(3\times10^{-1}-2.16\times10^{-1}=0.84\times10^{-1}=8.4\times10^{-2}\).

Sensor A's value is greater by \(8.4\times10^{-2}\).

- Identify the operation that combines a rate and elapsed time. - Work with coefficients and powers of \(10\) separately. - Check whether the coefficient needs to be normalized.

1. Multiply the rate by the time: \((3.2\times10^6)(4.5\times10^2)\). 2. Multiply coefficients and add exponents: \(14.4\times10^8\). 3. Normalize: \(14.4\times10^8=1.44\times10^9\).

\(1.44\times10^9\) bytes

- Use the operation associated with the area of a rectangle. - Include the squared unit in the result. - A coefficient at the boundary of the allowed range must be adjusted.

1. Multiply the dimensions: \((2.5\times10^{-3})(4\times10^{-4})\,\text{m}^2\). 2. The product is \(10\times10^{-7}\,\text{m}^2\). 3. Normalize: \(10\times10^{-7}=1\times10^{-6}\).

\(1\times10^{-6}\,\text{m}^2\)

- Combine the number of equal objects with the amount for one object. - Anticipate whether the total remains less than one gram. - Normalize after combining the factors.

1. Multiply the number of particles by the mass per particle. 2. \((6\times10^5)(2.5\times10^{-12})=15\times10^{-7}\,\text{g}\). 3. Normalize: \(15\times10^{-7}=1.5\times10^{-6}\).

\(1.5\times10^{-6}\,\text{g}\)

- First find the equal share represented by one drop. - Keep the original unit until the equal-share calculation is complete. - Apply the stated unit relationship to the final quantity.

1. Divide the total volume by the number of drops: \(\frac{9.6\times10^{-1}}{3.2\times10^4}=3\times10^{-5}\,\text{L}\). 2. Convert liters to milliliters: \((3\times10^{-5})(10^3)=3\times10^{-2}\,\text{mL}\).

\(3\times10^{-2}\,\text{mL}\) per drop

- Decide whether the total should be multiplied or divided by the capacity. - Estimate the number of pallets from the sizes of the powers of \(10\). - Check whether any partial pallet remains.

1. Divide the total number of tiles by the number per pallet. 2. \(\frac{7.5\times10^7}{2.5\times10^3}=3\times10^4\). 3. The quotient is a whole number, so exactly \(3\times10^4\) full pallets are needed.

\(3\times10^4\) pallets

- Use the relationship among speed, time, and distance. - Complete the distance calculation before changing units. - The stated unit conversion is also a power-of-ten operation.

1. Multiply speed by time: \((2.4\times10^2)(3\times10^2)=7.2\times10^4\,\text{cm}\). 2. Convert centimeters to meters: \(\frac{7.2\times10^4}{10^2}=7.2\times10^2\,\text{m}\).

\(7.2\times10^2\,\text{m}\)

- A prism's volume combines all three dimensions multiplicatively. - Track the unit through each factor. - Combine all factors before normalizing the coefficient.

1. Multiply all three dimensions: \((2\times10^2)(3\times10^{-1})(5\times10^{-3})\,\text{cm}^3\). 2. Multiply coefficients and add exponents: \(30\times10^{-2}\,\text{cm}^3\). 3. Normalize: \(30\times10^{-2}=3\times10^{-1}\).

\(3\times10^{-1}\,\text{cm}^3\)

- Identify whether the displayed operation allows powers to be combined directly. - Rewrite the quantities in a common place-value unit. - Use magnitude to check whether the proposed answer could be a sum.

1. Exponents are not added when the operation is addition; the place values must first match. 2. Rewrite \(3.2\times10^{-8}=0.32\times10^{-7}\). 3. Add: \((4.8+0.32)\times10^{-7}=5.12\times10^{-7}\).

Maya used the exponent rule for multiplication during addition. The correct sum is \(5.12\times10^{-7}\).

- Separate the problem into finding a total and then an equal share. - Keep the intermediate total in scientific notation. - Check whether the final count is reasonable compared with one cycle's collection.

1. Find the total collected: \((3.6\times10^6)(2.5\times10^2)=9\times10^8\) particles. 2. Divide among cartridges: \(\frac{9\times10^8}{9\times10^3}=1\times10^5\) particles per cartridge.

\(1\times10^5\) particles per cartridge

- Round the coefficients to nearby whole numbers before choosing a magnitude. - Combine the powers to identify the likely order of magnitude. - Compare the estimate with the normalized exact result.

1. Estimate \(8.1\approx8\) and \(2.9\approx3\), giving \(24\times10^1=2.4\times10^2\). Therefore \(2.3\times10^2\) is the reasonable estimate. 2. Compute exactly: \((8.1\cdot2.9)\times10^{5-4}=23.49\times10^1\). 3. Normalize: \(23.49\times10^1=2.349\times10^2\).

The reasonable estimate is \(2.3\times10^2\). The exact product is \(2.349\times10^2\).

- The quantities already use a common power-of-ten unit. - Find the total before calculating the equal share. - Check that the mean lies between the least and greatest values.

1. Add the file sizes: \((2.4+3.1+4.1)\times10^6=9.6\times10^6\) bytes. 2. Divide by \(3\): \(\frac{9.6\times10^6}{3}=3.2\times10^6\) bytes.

\(3.2\times10^6\) bytes

- Undo the subtraction to recover the starting quantity. - Align the powers of \(10\) before combining terms. - Substitute the result into the original equation to check it.

1. Add \(2.7\times10^5\) to the known difference: \(x=4.5\times10^6+2.7\times10^5\). 2. Rewrite \(2.7\times10^5=0.27\times10^6\). 3. Add: \(x=(4.5+0.27)\times10^6=4.77\times10^6\).

\(x=4.77\times10^6\)

- First turn the total and elapsed time into a rate. - Compare the actual rate with the stated target before finding a difference. - Normalize the final difference if its coefficient falls below \(1\).

1. Find the actual rate: \(\frac{6.4\times10^8}{8\times10^4}=0.8\times10^4=8\times10^3\) operations per minute. 2. Compare with the target: \(8\times10^3>7.5\times10^3\), so the target is met. 3. Find the excess: \((8-7.5)\times10^3=0.5\times10^3=5\times10^2\) operations per minute.

Yes. The machine exceeds the target by \(5\times10^2\) operations per minute.

- Convert the standard-form dividend before dividing. - Treat the coefficients and powers as separate parts. - Normalize a quotient whose coefficient is less than \(1\).

1. Convert \(72{,}000{,}000=7.2\times10^7\). 2. Divide: \(\frac{7.2\times10^7}{9\times10^3}=0.8\times10^4\). 3. Normalize: \(0.8\times10^4=8\times10^3\).

\(8\times10^3\)

- Find the gap between the limit and the current measurement. - Express both quantities using the smaller exponent's place-value unit. - Confirm that the gap plus the current value returns to the limit.

1. Rewrite the limit as \(1\times10^{-4}=10\times10^{-5}\). 2. Subtract: \((10-6.75)\times10^{-5}=3.25\times10^{-5}\,\text{m}\).

\(3.25\times10^{-5}\,\text{m}\)

- First determine the total produced over all days. - Put the removal amount in the same power-of-ten unit as the total. - Check that the remaining count is less than the produced count.

1. Find the number printed: \((1.8\times10^5)(2\times10^1)=3.6\times10^6\) labels. 2. Rewrite the removed amount: \(4\times10^5=0.4\times10^6\). 3. Subtract: \((3.6-0.4)\times10^6=3.2\times10^6\) labels.

\(3.2\times10^6\) labels

- Complete each operation box before moving to the one above it. - Carry each intermediate value forward in normalized form. - Check that each division makes the value smaller.

1. a) Add: \(2.4\times10^7+3.6\times10^7=6\times10^7\). 2. b) Divide by \(8\): \(\frac{6\times10^7}{8}=0.75\times10^7=7.5\times10^6\). 3. c) Divide by \(3\): \(\frac{7.5\times10^6}{3}=2.5\times10^6\).

a) \(6\times10^7\) b) \(7.5\times10^6\) c) \(2.5\times10^6\)

- Check whether both coefficients currently refer to the same power-of-ten unit. - Rewrite one addend without changing its value before adding. - Estimate whether the correct result should be only a little larger than the larger addend.

1. The coefficients cannot be added directly because the two powers of \(10\) represent different place values. 2. Rewrite \(3.1\times10^4=0.31\times10^5\). 3. Add the aligned coefficients: \((4.2+0.31)\times10^5=4.51\times10^5\).

Noah added coefficients that referred to different place values. The correct sum is \(4.51\times10^5\).

- Convert each actual measurement to the unit you want for the model. - Divide every actual length by \(10^9\). - Use exponent rules to simplify the powers of ten.

1. Using the approximate diameter of the Sun, \(1.4\times 10^6\,\text{km}=1.4\times 10^9\,\text{m}\). Dividing the rounded value by \(10^9\) gives a model diameter of approximately \(1.4\,\text{m}\). 2. Using the approximate diameter of Earth, \(1.28\times 10^4\,\text{km}=1.28\times 10^{10}\,\text{mm}\). Dividing the rounded value by \(10^9\) gives a model diameter of approximately \(12.8\,\text{mm}\). 3. Using the approximate Earth-Sun distance, \(1.5\times 10^8\,\text{km}=1.5\times 10^{11}\,\text{m}\). Dividing the rounded value by \(10^9\) gives a model distance of approximately \(150\,\text{m}\).

a) Approximately \(1.4\,\text{m}\) b) Approximately \(12.8\,\text{mm}\) c) Approximately \(150\,\text{m}\)

- Multiply the light-year value by \(4.2\). - Divide the actual distance by the kilometers represented by one model centimeter. - Convert centimeters to meters and kilometers.

1. Using the rounded light-year value, the estimated actual distance is \(4.2\cdot(1.0\times 10^{13})\approx 4.2\times 10^{13}\,\text{km}\), or approximately \(42{,}000{,}000{,}000{,}000\,\text{km}\). 2. The model length is approximately \((4.2\times 10^{13})\div(1.0\times 10^8)=4.2\times 10^5\,\text{cm}\). 3. Convert the model length: \(4.2\times 10^5\,\text{cm}=4200\,\text{m}=4.2\,\text{km}\).

a) Approximately \(42{,}000{,}000{,}000{,}000\,\text{km}\) b) Approximately \(4.2\,\text{km}\)

- Estimate how many equal contributions are needed to reach the threshold. - Test the whole numbers immediately below and above that estimate. - The word “exceed” rules out a total that is merely below or equal to the threshold.

1. Eleven scans give \(11(3.6\times10^{-8})=39.6\times10^{-8}=3.96\times10^{-7}\), which does not exceed the threshold. 2. Twelve scans give \(12(3.6\times10^{-8})=43.2\times10^{-8}=4.32\times10^{-7}\), which exceeds the threshold. 3. Therefore the least number is \(12\).

\(12\) scans, producing \(4.32\times10^{-7}\) units of signal

- For the estimate, round the coefficients before combining the powers of \(10\). - For the exact calculation, align the powers of \(10\) before adding the two numerator terms. - After the final division, compare the size category of the exact result with your rough estimate.

1. a) For a rough estimate, use \(4.8\times10^7\approx5\times10^7\) and \(2.5\times10^{-3}\approx3\times10^{-3}\). Their product is about \(1.5\times10^5\). Adding \(3\times10^4\) gives about \(1.8\times10^5\). Dividing by \(6\times10^2\) gives about \(3\times10^2\), so the result should be in the hundreds. 2. b) Multiply first: \((4.8\times10^7)\cdot(2.5\times10^{-3})=12\times10^4=1.2\times10^5\). 3. Align powers for the addition: \(3.0\times10^4=0.30\times10^5\), so the numerator is \(1.50\times10^5\). 4. Divide: \(\frac{1.50\times10^5}{6\times10^2}=0.25\times10^3=2.5\times10^2\). 5. c) The exact result, \(2.5\times10^2\), is in the hundreds, which matches the rough estimate.

a) Hundreds; a rough estimate is \(\frac{1.8\times10^5}{6\times10^2}\approx3\times10^2\). b) \(2.5\times10^2\) c) The exact result is in the hundreds, matching the estimate.