Scientific Notation Flashcards: Conversions, Operations & Error Checks

Question 1

For a nonzero number in normalized scientific notation a	imes10^n, what restrictions apply to a and n?

Accepted Answer

1\leq |a|<10, and n is an integer.

The coefficient may be negative; its absolute value must be at least 1 and less than 10.

Question 2

Exact arithmetic: write 680000 in normalized scientific notation.

Accepted Answer

6.8	imes10^5

The decimal moves 5 places left; the factor 10^5 preserves the value.

Question 3

Exact arithmetic: evaluate 10^0.

Accepted Answer

1

A zero exponent does not make a nonzero base equal zero.

Question 4

Exact arithmetic: write 7.3	imes10^{-4} as an ordinary decimal.

Accepted Answer

0.00073

Multiplying by 10^{-4} divides 7.3 by 10000.

Question 5

Exact arithmetic: normalize 48.2	imes10^3 without changing its value.

Accepted Answer

4.82	imes10^4

Dividing the coefficient by 10 requires multiplying the power-of-ten factor by 10.

Question 6

Exact arithmetic: write 0.0000916 in normalized scientific notation.

Accepted Answer

9.16	imes10^{-5}

The decimal moves 5 places right to make the coefficient; the exponent is negative.

Question 7

Exact arithmetic: evaluate 10^{-4} as an ordinary decimal.

Accepted Answer

0.0001

The negative exponent means a reciprocal: 10^{-4}=1/10^4. The value is positive.

Question 8

Exact arithmetic: write 2.064	imes10^6 as an ordinary decimal.

Accepted Answer

2064000

Move the decimal 6 places right.

Question 9

Exact arithmetic: normalize 0.063	imes10^5 without changing its value.

Accepted Answer

6.3	imes10^3

The coefficient increases by a factor of 100, so the power-of-ten factor decreases by a factor of 100.

Question 10

Exact arithmetic: write -0.00408 in normalized scientific notation.

Accepted Answer

-4.08	imes10^{-3}

Keep the negative sign on the coefficient. The exponent describes the decimal scale.

Question 11

Exact arithmetic: write -8.2	imes10^2 as an ordinary decimal.

Accepted Answer

-820

Multiplying by 100 changes the size, while the negative sign remains.

Question 12

Exact arithmetic: write 5.04 in normalized scientific notation.

Accepted Answer

5.04	imes10^0

The coefficient is already in the required range, and 10^0=1.

Question 13

Exact arithmetic: normalize -720	imes10^{-5} without changing its value.

Accepted Answer

-7.2	imes10^{-3}

Divide the coefficient by 100 and increase the exponent by 2.

Question 14

Exact arithmetic: write -3.15	imes10^{-2} as an ordinary decimal.

Accepted Answer

-0.0315

Divide 3.15 by 100 and keep the negative sign.

Question 15

Exact arithmetic: calculate (2.4	imes10^3)(3	imes10^4). Give normalized scientific notation.

Accepted Answer

7.2	imes10^7

Multiply the coefficients: 2.4	imes3=7.2. Add the exponents: 3+4=7.

Question 16

Can zero satisfy the normalized scientific-notation condition 1\leq|a|<10?

Accepted Answer

No. Write zero as 0.

A coefficient satisfying that condition cannot produce zero when multiplied by a power of ten.

Question 17

Exact arithmetic: normalize 0.0056	imes10^{-2} without changing its value.

Accepted Answer

5.6	imes10^{-5}

Multiply the coefficient by 1000 and decrease the exponent by 3.

Question 18

Exact arithmetic: calculate (8.1	imes10^8)/(2.7	imes10^3). Give normalized scientific notation.

Accepted Answer

3	imes10^5

Divide the coefficients and subtract the denominator exponent: 8-3=5.

Question 19

Exact arithmetic: calculate (3.16	imes10^4)+(2.42	imes10^4). Give normalized scientific notation.

Accepted Answer

5.58	imes10^4

The powers match, so add the coefficients and retain 10^4.

Question 20

Exact arithmetic: calculate (6.4	imes10^{-3})(5	imes10^2). Give normalized scientific notation.

Accepted Answer

3.2	imes10^0

The product is 32	imes10^{-1}. Normalize by dividing 32 by 10 and increasing the exponent by 1.

Question 21

Exact arithmetic: calculate (7.85	imes10^{-6})-(2.65	imes10^{-6}). Give normalized scientific notation.

Accepted Answer

5.2	imes10^{-6}

The powers match; subtract the coefficients: 7.85-2.65=5.2.

Question 22

Exact arithmetic: normalize 10	imes10^{-7} without changing its value.

Accepted Answer

1	imes10^{-6}

A coefficient of 10 is outside the normalized range. Divide it by 10 and increase the exponent by 1.

Question 23

Exact arithmetic: calculate (1.8	imes10^5)/(6	imes10^2). Give normalized scientific notation.

Accepted Answer

3	imes10^2

The quotient is 0.3	imes10^3. Multiply the coefficient by 10 and decrease the exponent by 1.

Question 24

Exact arithmetic: calculate (4.2	imes10^5)+(3.6	imes10^4). Give normalized scientific notation.

Accepted Answer

4.56	imes10^5

Rewrite 3.6	imes10^4 as 0.36	imes10^5, then add 4.2+0.36.

Question 25

Exact arithmetic: repair the exponent: 0.00052=5.2	imes10^4.

Accepted Answer

0.00052=5.2	imes10^{-4}

The number is smaller than 1. A positive exponent here would give 52000.

Question 26

Exact arithmetic: calculate (-1.6	imes10^{-4})(2.5	imes10^{-2}). Give normalized scientific notation.

Accepted Answer

-4	imes10^{-6}

The signs give a negative product; 1.6	imes2.5=4 and -4+(-2)=-6.

Question 27

Exact arithmetic: calculate (6.3	imes10^3)-(8	imes10^2). Give normalized scientific notation.

Accepted Answer

5.5	imes10^3

Rewrite 8	imes10^2 as 0.8	imes10^3, then subtract 6.3-0.8.

Question 28

Exact arithmetic: calculate (7.2	imes10^{-5})/(2.4	imes10^{-2}). Give normalized scientific notation.

Accepted Answer

3	imes10^{-3}

Divide the coefficients: 7.2/2.4=3. The exponent is -5-(-2)=-3.

Question 29

Exact arithmetic: calculate (7.8	imes10^6)+(6.5	imes10^6). Give normalized scientific notation.

Accepted Answer

1.43	imes10^7

The sum is 14.3	imes10^6; normalize the coefficient by moving one factor of 10 into the power.

Question 30

Exact arithmetic: normalize -0.92	imes10^0 without changing its value.

Accepted Answer

-9.2	imes10^{-1}

Increase the coefficient magnitude by a factor of 10 and decrease the exponent by 1.

Question 31

Exact arithmetic: calculate (4	imes10^{-7})(1.5	imes10^7). Give normalized scientific notation.

Accepted Answer

6	imes10^0

The coefficient product is 6. The exponents add to zero, so the power-of-ten factor is 1.

Question 32

Exact arithmetic: calculate (2.1	imes10^{-3})-(8.4	imes10^{-3}). Give normalized scientific notation.

Accepted Answer

-6.3	imes10^{-3}

The powers match, and 2.1-8.4=-6.3. A negative coefficient is allowed.

Question 33

Exact arithmetic: calculate (6.3	imes10^{-2})/(-9	imes10^{-4}). Give normalized scientific notation.

Accepted Answer

-7	imes10^1

The quotient is -0.7	imes10^2 because -2-(-4)=2. Normalize to -7	imes10^1.

Question 34

Exact arithmetic: repair the final expression: 37	imes10^4=3.7	imes10^3.

Accepted Answer

37	imes10^4=3.7	imes10^5

Dividing the coefficient by 10 must be offset by increasing the exponent by 1.

Question 35

Exact arithmetic: calculate (-5.2	imes10^{-4})+(1.7	imes10^{-3}). Give normalized scientific notation.

Accepted Answer

1.18	imes10^{-3}

Use a common power: -0.52	imes10^{-3}+1.7	imes10^{-3}. Add the signed coefficients.

Question 36

Exact arithmetic: calculate (-3.5	imes10^2)(-2	imes10^{-5}). Give normalized scientific notation.

Accepted Answer

7	imes10^{-3}

Two negative factors give a positive product. Multiply 3.5	imes2 and add 2+(-5).

Question 37

Exact arithmetic: calculate (4.8	imes10^3)/(1.2	imes10^{-4}). Give normalized scientific notation.

Accepted Answer

4	imes10^7

The coefficient quotient is 4. Subtract the negative denominator exponent: 3-(-4)=7.

Question 38

Exact arithmetic: calculate (4.6	imes10^9)-(4.6	imes10^9).

Accepted Answer

0

Identical values cancel exactly. Zero has no representation satisfying 1\leq|a|<10.

Question 39

Exact arithmetic: repair the result: (3	imes10^4)+(2	imes10^3)=5	imes10^7.

Accepted Answer

3.2	imes10^4

Rewrite the second term as 0.2	imes10^4 and add 3+0.2. Adding exponents is a multiplication rule, not an addition rule.

Question 40

Exact arithmetic: calculate (8	imes10^{-6})(7	imes10^{-3}). Give normalized scientific notation.

Accepted Answer

5.6	imes10^{-8}

The product is 56	imes10^{-9}. Dividing the coefficient by 10 raises the exponent to -8.

Question 41

Exact arithmetic: calculate (2.1	imes10^{-5})/(7	imes10^{-5}). Give normalized scientific notation.

Accepted Answer

3	imes10^{-1}

Equal powers cancel, leaving 0.3	imes10^0. Normalize the coefficient to 3 and decrease the exponent by 1.

Question 42

Exact arithmetic: repair the exponent: (9	imes10^2)/(3	imes10^5)=3	imes10^3.

Accepted Answer

3	imes10^{-3}

Division subtracts the denominator exponent from the numerator exponent: 2-5=-3.

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Scientific Notation Flashcards: Conversions, Operations & Error Checks