Scientific Notation: Practice

Multiply the coefficients and add the exponents: $5 \times 6 = 30$ and $4 + 2 = 6$, giving $30 \times 10^{6}$. The coefficient $30$ is $10$ or more, so renormalize by writing $30 = 3 \times 10$.

$30 \times 10^{6} = 3 \times 10^{1} \times 10^{6} = 3 \times 10^{7}$

The proper form is $3 \times 10^{7}$.

Divide the coefficients and subtract the exponents.

$\frac{9.6 \times 10^{8}}{3 \times 10^{5}} = \frac{9.6}{3} \times 10^{8-5} = 3.2 \times 10^{3}$

The coefficient $3.2$ is in range, so the answer is $3.2 \times 10^{3}$.

Multiply the coefficients and add the exponents: $8 \times 5 = 40$ and $3 + 6 = 9$, giving $40 \times 10^{9}$. Renormalize, since $40 = 4 \times 10$.

$40 \times 10^{9} = 4 \times 10^{1} \times 10^{9} = 4 \times 10^{10}$

The proper form is $4 \times 10^{10}$.

Divide the coefficients and subtract the exponents: $\frac{1.5}{5} = 0.3$ and $4 - 6 = -2$, giving $0.3 \times 10^{-2}$. The coefficient $0.3$ is below $1$, so renormalize by writing $0.3 = 3 \times 10^{-1}$.

$0.3 \times 10^{-2} = 3 \times 10^{-1} \times 10^{-2} = 3 \times 10^{-3}$

Borrowing a factor of ten lowers the exponent by $1$.

Distance is speed times time, so multiply the two quantities: multiply the coefficients and add the exponents.

$(3 \times 10^{5}) \times (2 \times 10^{2}) = 6 \times 10^{5+2} = 6 \times 10^{7}$

The coefficient $6$ is in range, so the distance is about $6 \times 10^{7}$ km.

Both coefficients are between $1$ and $10$, so the larger exponent gives the larger value. Compare $-27$ and $-31$.

$-27 > -31 \;\Rightarrow\; 1.7 \times 10^{-27} > 9.1 \times 10^{-31}$

The proton, with exponent $-27$, has more mass, even though its coefficient $1.7$ is smaller.

Multiply the coefficients and add the exponents: $2.5 \times 4 = 10$ and $6 + (-2) = 4$, giving $10 \times 10^{4}$. The coefficient $10$ is not less than $10$, so renormalize: $10 = 1 \times 10$.

$10 \times 10^{4} = 1 \times 10^{1} \times 10^{4} = 1 \times 10^{5}$

The proper form is $1 \times 10^{5}$.

The number is below $1$, so the point moves right for a negative exponent. Place it after the first nonzero digit, the $9$, and count the hops from $0.000000094$: that is $8$ places right.

$0.000000094 = 9.4 \times 10^{-8}$

Count the leading zeros carefully; the exponent is $-8$, not $-9$.

The powers of ten match, so this is like adding $7$ and $2$ of the same unit ($10^{4}$). Add the coefficients and keep the shared power of ten.

$(7 \times 10^{4}) + (2 \times 10^{4}) = (7 + 2) \times 10^{4} = 9 \times 10^{4}$

The exponent does not change when adding numbers with the same power of ten; only multiplying and dividing change it.

Dollars per person is the budget divided by the population: divide the coefficients and subtract the exponents.

$\frac{4 \times 10^{12}}{4 \times 10^{8}} = \frac{4}{4} \times 10^{12-8} = 1 \times 10^{4}$

That is about $10{,}000$ dollars per person.

Divide the Sun's diameter by Earth's: divide the coefficients and subtract the exponents.

$\frac{1.4 \times 10^{6}}{1.3 \times 10^{4}} \approx 1.08 \times 10^{6-4} = 1.08 \times 10^{2}$

That is about $108$, so the Sun is roughly $100$ times wider.

Divide the coefficients and subtract the exponents: $\frac{4.5}{9} = 0.5$ and $-2 - 3 = -5$, giving $0.5 \times 10^{-5}$. The coefficient $0.5$ is below $1$, so renormalize: $0.5 = 5 \times 10^{-1}$.

$0.5 \times 10^{-5} = 5 \times 10^{-1} \times 10^{-5} = 5 \times 10^{-6}$

The proper form is $5 \times 10^{-6}$.