Free math worksheets

- Place the decimal after the first nonzero digit. - Count how many positions the decimal point moves. - Use a positive exponent for a number greater than \(1\).

1. Move the decimal point so the coefficient is \(6.42\). 2. The decimal point moves \(6\) places to the left. 3. Therefore, the number is \(6.42\times10^6\).

\(6.42\times10^6\)

- Find the first nonzero digit. - Count the places needed to make a coefficient between \(1\) and \(10\). - Relate the exponent’s sign to the size of the original number.

1. Move the decimal point to form the coefficient \(8.3\). 2. The decimal point moves \(5\) places to the right. 3. The original number is less than \(1\), so the exponent is negative. 4. Therefore, the number is \(8.3\times10^{-5}\).

\(8.3\times10^{-5}\)

- Count from the original decimal point to the new one. - Use the size of the number to determine the exponent’s sign. - Substitute the exponent to check the equality.

1. Move the decimal point from \(0.00000456\) to make \(4.56\). 2. The decimal moves \(6\) places to the right. 3. Therefore, the exponent is \(-6\).

\(-6\)

- Place the decimal to create a valid coefficient. - Confirm that the stated power of \(10\) matches the original place value. - Read the coefficient from the corrected form.

1. Write \(32{,}000{,}000\) with the decimal after the first nonzero digit: \(3.2\). 2. The decimal point moves \(7\) places left. 3. Thus, \(32{,}000{,}000=3.2\times10^7\), so \(a=3.2\).

\(a=3.2\)

- Compare the coefficient directly with the target decimal. - Count the required place-value shift. - Use the direction of the shift to determine the sign.

1. Moving the decimal point in \(6.04\) three places left gives \(0.00604\). 2. A three-place move left corresponds to multiplication by \(10^{-3}\). 3. Therefore, \(n=-3\).

\(n=-3\)

- Express each metric prefix as a power of \(10\). - Move the decimal point until the coefficient is at least \(1\) and less than \(10\). - Adjust the exponent to compensate for the decimal shift.

1. Since \(1\,\text{mm}=10^{-3}\,\text{m}\), \(0.25\,\text{mm}=0.25\times10^{-3}\,\text{m}=2.5\times10^{-4}\,\text{m}\). 2. Since \(1\,\mu\text{m}=10^{-6}\,\text{m}\), \(2\,\mu\text{m}=2\times10^{-6}\,\text{m}\). 3. Since \(1\,\text{nm}=10^{-9}\,\text{m}\), \(25\,\text{nm}=25\times10^{-9}\,\text{m}=2.5\times10^{-8}\,\text{m}\).

a) \(2.5\times10^{-4}\,\text{m}\) b) \(2\times10^{-6}\,\text{m}\) c) \(2.5\times10^{-8}\,\text{m}\)

- Use the exponent to determine the direction and distance of the move. - Insert zeros as placeholders when needed. - Check whether the final value should be very large or very small.

1. For part a), move the decimal point \(8\) places right: \(407{,}000{,}000\). 2. For part b), move the decimal point \(6\) places left: \(0.0000092\).

a) \(407{,}000{,}000\) b) \(0.0000092\)

- Adjust each coefficient into the required interval. - Compensate for every coefficient shift by changing the exponent. - Check that the rewritten value has not changed.

1. \(12.5\times10^4=1.25\times10^5\). 2. \(0.63\times10^{-2}=6.3\times10^{-3}\). 3. \(84\times10^0=8.4\times10^1\). 4. Each corrected coefficient is at least \(1\) and less than \(10\).

a) \(1.25\times10^5\) b) \(6.3\times10^{-3}\) c) \(8.4\times10^1\)

- Interpret “E” as multiplication by a power of \(10\). - Use the signed exponent to move the decimal point. - Count placeholder zeros carefully.

1. The calculator notation \(5.806\text{E}-7\) means \(5.806\times10^{-7}\). 2. Moving the decimal point \(7\) places left gives \(0.0000005806\).

a) \(5.806\times10^{-7}\) b) \(0.0000005806\)

- Compare the powers of \(10\) before the coefficients. - Decide whether the coefficients need to be compared at all. - Check the result against the approximate sizes of the numbers.

1. The exponents are \(5\) and \(6\). 2. In normalized scientific notation, \(7.1\times10^5<10\times10^5=10^6\), while \(6.9\times10^6>10^6\). 3. Therefore, \(7.1\times10^5<6.9\times10^6\). The exponents decide the comparison before the coefficients.

\(7.1\times10^5<6.9\times10^6\)

- First identify the number’s order of magnitude. - Round the coefficient to one nonzero digit. - Keep the power of \(10\) consistent with the rounded coefficient.

1. \(783{,}000{,}000=7.83\times10^8\). 2. Rounding the coefficient \(7.83\) to one digit gives \(8\). 3. Therefore, the estimate is \(8\times10^8\).

\(8\times10^8\)

- Convert the small decimal to normalized scientific notation. - Round only the coefficient. - Preserve the correct order of magnitude.

1. Write the exact number as \(6.72\times10^{-6}\). 2. Round the coefficient \(6.72\) to one digit: \(7\). 3. The estimate is \(7\times10^{-6}\).

\(7\times10^{-6}\)

- Check each coefficient against the required interval. - Shift the coefficient by one decimal place when needed. - Change the exponent in the opposite way to preserve the value.

1. Part a) is valid because \(1\le9.99<10\). 2. Part b) is invalid; \(10\times10^2=1\times10^3\). 3. Part c) is invalid; \(0.8\times10^5=8\times10^4\). 4. Part d) is valid because \(5\) is an allowed coefficient.

a) Valid b) \(1\times10^3\) c) \(8\times10^4\) d) Valid

- Use the size of the original number to predict the exponent’s sign. - Count the decimal-place movement independently. - Substitute the corrected power of \(10\) to check the value.

1. The original number is less than \(1\), so its scientific-notation exponent must be negative. 2. Moving the decimal point from \(0.00072\) to \(7.2\) requires \(4\) places. 3. Therefore, \(0.00072=7.2\times10^{-4}\).

\(0.00072=7.2\times10^{-4}\)

- First convert the magnitude of the number. - Keep the coefficient’s absolute value between \(1\) and \(10\). - Restore the original sign after determining the exponent.

1. Ignore the sign temporarily and write \(0.00091=9.1\times10^{-4}\). 2. Restore the negative sign to the coefficient. 3. Therefore, \(-0.00091=-9.1\times10^{-4}\).

\(-9.1\times10^{-4}\)

- Preserve the zero already inside the coefficient. - Add placeholder zeros until the decimal has moved the required number of places. - Count zeros only after writing the complete numeral.

1. Move the decimal point \(12\) places to the right. 2. The standard form is \(3{,}040{,}000{,}000{,}000\). 3. This numeral contains \(11\) zeros: one between \(3\) and \(4\), and ten after the \(4\).

\(3{,}040{,}000{,}000{,}000\), with \(11\) zeros

- Convert the decimal to the same representation as the other number. - Once exponents match, compare coefficients. - Check the conclusion by reading the first differing decimal place.

1. Convert \(0.000061=6.1\times10^{-5}\). 2. The exponents match, so compare coefficients. 3. Since \(6.2>6.1\), \(6.2\times10^{-5}>0.000061\).

\(6.2\times10^{-5}>0.000061\)

- Read the sign attached to the calculator exponent carefully. - Translate E-notation before moving any decimal point. - Check whether the displayed sign should produce a large or small number.

1. The display \(\text{E}+05\) means multiply by \(10^5\), not \(10^{-5}\). 2. The correct scientific notation is \(2.04\times10^5\). 3. Moving the decimal point \(5\) places right gives \(204{,}000\).

\(2.04\times10^5=204{,}000\)

- Normalize the coefficient first. - Count how many places the coefficient changes. - Compensate with the exponent in the opposite direction.

1. Move the coefficient’s decimal point \(2\) places right: \(0.072\) becomes \(7.2\). 2. To preserve the value, decrease the exponent by \(2\). 3. Therefore, \(0.072\times10^9=7.2\times10^7\).

\(7.2\times10^7\)

- Track each digit of the coefficient by place value. - Insert placeholder zeros without deleting existing ones. - Compare the expanded number with the original coefficient structure.

1. Move the decimal point \(9\) places to the right. 2. The standard form is \(7{,}005{,}000{,}000\). 3. The zeros in \(7.005\) hold the hundred-millions and ten-millions places, so removing them would change the value.

\(7{,}005{,}000{,}000\)

- Move every digit of the coefficient together. - Use zeros as place-value holders. - Locate the final place of the last nonzero digit.

1. Move the decimal point \(6\) places to the right. 2. The result is \(1{,}000{,}500\). 3. The digit \(5\) lands in the hundreds place because it began four places to the right of the coefficient’s ones digit.

\(1{,}000{,}500\)

- Match the power of \(10\) to a familiar metric prefix. - Choose a unit that absorbs the entire power-of-ten factor. - Keep the coefficient unchanged.

1. Since \(1\,\text{mm}=10^{-3}\,\text{m}\), the factor \(10^{-3}\,\text{m}\) corresponds to one millimeter. 2. Therefore, \(2.5\times10^{-3}\,\text{m}=2.5\,\text{mm}\).

\(2.5\,\text{mm}\)

- Separate numerical equality from notation requirements. - Check the coefficient range. - Renormalize without changing the value.

1. The equality \(45{,}000{,}000=45\times10^6\) is numerically correct. 2. The coefficient \(45\) is not less than \(10\), so the expression is not normalized scientific notation. 3. Move the coefficient’s decimal one place left and increase the exponent by \(1\). 4. The normalized form is \(4.5\times10^7\).

The equality is correct, but the normalized scientific notation is \(4.5\times10^7\).

- Check which coefficient is outside the normalized interval. - Shift that coefficient one place. - Compensate with the exponent and compare the results.

1. The first expression is already normalized. 2. Normalize the second: \(74\times10^{-10}=7.4\times10^{-9}\). 3. Therefore, the expressions are equal.

They are equal; both normalize to \(7.4\times10^{-9}\).

- Compare the coefficients before comparing the powers of \(10\). - A multiplicative comparison can be represented by a quotient. - Think about what the difference between the two exponents represents.

1. The coefficients are equal, so the scale factor comes entirely from the powers of \(10\). 2. \(\frac{6.2\times10^7}{6.2\times10^3}=10^{7-3}=10^4\). 3. Since \(10^4=10{,}000\), the first number is \(10{,}000\) times as large as the second.

The exponent difference is \(7-3=4\), so the scale factor is \(10^4=10{,}000\). The first number is \(10{,}000\) times as large as the second.

- 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) \((7.1\times10^0)\cdot10^3=7.1\times10^3\,\text{cm}^3\); \(\frac{7.1\times10^0}{10^3}=7.1\times10^{-3}\,\text{m}^3\)

- For negative exponents, identify which power of \(10\) gives the smaller place value. - Group numbers with equal exponents. - Compare coefficients only within the same power of \(10\).

1. \(8.1\times10^{-4}=0.00081\), while the other two numbers are in the thousandths. 2. Thus, \(8.1\times10^{-4}\) is least. 3. The remaining exponents match, so compare coefficients: \(3.9<4.2\). 4. Therefore, \(8.1\times10^{-4}<3.9\times10^{-3}<4.2\times10^{-3}\).

\(8.1\times10^{-4}<3.9\times10^{-3}<4.2\times10^{-3}\)

- Use the power-of-ten relationship between meters and millimeters. - Track how the unit change affects the numerical value. - Normalize the final scientific notation.

1. Since \(1\,\text{m}=10^3\,\text{mm}\), multiply the meter value by \(10^3\). 2. \(7.2\times10^{-4}\,\text{m}=7.2\times10^{-1}\,\text{mm}\). 3. In decimal form, this is \(0.72\,\text{mm}\).

\(0.72\,\text{mm}=7.2\times10^{-1}\,\text{mm}\)

- Consider whether either factor in the required form can equal zero. - Use the coefficient restriction explicitly. - Decide whether any integer exponent changes that conclusion.

1. Every power \(10^n\) is nonzero. 2. If \(1\le a<10\), then \(a\) is also nonzero. 3. The product \(a\times10^n\) is therefore nonzero. 4. Thus, no normalized scientific-notation expression with the stated coefficient condition equals \(0\).

No. Zero is written simply as \(0\); it has no normalized scientific-notation form with \(1\le a<10\).

- Translate the named place value into a power of \(10\). - Multiply that unit by the stated count. - Normalize the coefficient and verify with the decimal form.

1. One billionth is \(10^{-9}\). 2. Therefore, \(43\) billionths is \(43\times10^{-9}\). 3. Normalize the coefficient: \(43\times10^{-9}=4.3\times10^{-8}\). 4. The decimal form is \(0.000000043\).

Decimal: \(0.000000043\) Scientific notation: \(4.3\times10^{-8}\)

- Use the interval between consecutive powers of \(10\). - Normalize the coefficient before selecting the exponent. - Test whether the alternative exponent would produce an allowed coefficient.

1. Place the decimal after the first nonzero digit: \(8.4\). 2. Moving the decimal point \(4\) places left gives \(8.4\times10^4\). 3. The exponent is \(4\) because the normalized coefficient \(8.4\) multiplies \(10^4\); using \(10^5\) would require coefficient \(0.84\), which is not normalized.

\(84{,}000=8.4\times10^4\)

- Keep the exponent fixed and focus on the allowed coefficient interval. - Choose a coefficient that meets the decimal-place condition. - Expand the power of \(10\) to verify both inequalities.

1. One valid coefficient between \(7\) and \(8\) with two decimal places is \(7.25\). 2. A valid number is \(7.25\times10^{-6}\). 3. Its decimal form is \(0.00000725\). 4. Any coefficient from \(7.01\) through \(7.99\) with exactly two decimal places gives another valid answer.

One example is \(7.25\times10^{-6}=0.00000725\).

- Write the exact normalized form first. - Notice what happens when the coefficient rounds to \(10\). - Renormalize the rounded result.

1. \(999{,}000=9.99\times10^5\). 2. Rounding \(9.99\) to one digit gives \(10\), so the intermediate form is \(10\times10^5\). 3. Normalize: \(10\times10^5=1\times10^6\). 4. The exponent increases because the rounded coefficient reaches \(10\).

\(1\times10^6\)

- Express the unit conversion as a power of \(10\). - Combine the powers while preserving the coefficient. - Normalize and then write the ordinary decimal form.

1. Since \(1\,\text{km}=10^6\,\text{mm}\), multiply by \(10^6\). 2. \(4.8\times10^{-6}\,\text{km}=4.8\times10^0\,\text{mm}\). 3. Therefore, the standard form is \(4.8\,\text{mm}\).

\(4.8\,\text{mm}=4.8\times10^0\,\text{mm}\)

- Rewrite one number using the other number’s power of \(10\). - Compare coefficients only after the exponents match. - Pay attention to values near a power-of-ten boundary.

1. Rewrite \(9.9\times10^3\) with exponent \(4\): \(0.99\times10^4\). 2. Compare coefficients \(0.99\) and \(1.01\). 3. Since \(0.99<1.01\), \(9.9\times10^3<1.01\times10^4\).

\(9.9\times10^3<1.01\times10^4\)

- Compare the negative exponents before the coefficients. - Among equal exponents, compare coefficients normally. - Expand only the greatest and least readings after identifying them.

1. Sensor B has exponent \(-6\), so its reading is greater than the readings with exponent \(-7\). 2. Between A and C, the exponents match and \(7.5<8.1\), so A is least. 3. The greatest reading is \(6.9\times10^{-6}=0.0000069\). 4. The least reading is \(7.5\times10^{-7}=0.00000075\).

Greatest: Sensor B, \(0.0000069\) Least: Sensor A, \(0.00000075\)

- Construct the coefficient before applying the exponent. - Check both the digit-sum and decimal-place conditions. - Expand the completed scientific notation to verify it.

1. One valid coefficient is \(3.45\), whose digits satisfy \(3+4+5=12\). 2. The scientific-notation number is \(3.45\times10^{-5}\). 3. Its standard form is \(0.0000345\). 4. Other normalized coefficients with two decimal places and digit sum \(12\) are also valid.

One example is \(3.45\times10^{-5}=0.0000345\).

- Convert the decimal reading first. - Normalize the reading whose coefficient is too large. - Compare only after every reading has the same standard form.

1. Reading A is \(0.00000048=4.8\times10^{-7}\). 2. Reading B is already \(4.8\times10^{-7}\). 3. Reading C normalizes as \(48\times10^{-8}=4.8\times10^{-7}\). 4. Therefore, all three readings are equal.

A, B, and C all equal \(4.8\times10^{-7}\).

- Express the stated multiplicative factor as a power of \(10\). - Because the coefficients match, focus on the difference between the exponents. - Relate that exponent difference to the given scale factor.

1. Write the scale factor as a power of \(10\): \(1{,}000{,}000=10^6\). 2. The coefficients of \(A\) and \(B\) are equal, so \(\frac{A}{B}=10^{n-(-3)}=10^{n+3}\). 3. Set the exponent difference equal to \(6\): \(n+3=6\). 4. Therefore, \(n=3\).

\(n=3\)