Scientific Notation: Additional Practice (Set 2)

Multiply the coefficients and add the exponents.

$(4 \times 10^{5}) \times (2 \times 10^{3}) = (4 \times 2) \times 10^{5+3} = 8 \times 10^{8}$

The coefficient $8$ is already between $1$ and $10$, so no renormalizing is needed.

Order by exponent first. The exponents are $6, 5, 5, 4$, so $2 \times 10^{6}$ leads and $3 \times 10^{4}$ trails. The two with exponent $5$ are split by their coefficients: $9 > 8$.

$2 \times 10^{6} > 9 \times 10^{5} > 8 \times 10^{5} > 3 \times 10^{4}$

Convert $2.4 \times 10^{5} = 240{,}000$, then test each option. $24 \times 10^{4} = 24 \times 10{,}000 = 240{,}000$.

$2.4 \times 10^{5} = 240{,}000 = 24 \times 10^{4}$

The others give $2{,}400{,}000$, $24{,}000$, and $2{,}400$, none of which equal $240{,}000$.

Convert each to an ordinary number. The first three give $36{,}000$, but check the last one.

$3.6 \times 10^{3} = 3{,}600$

That is $3{,}600$, not $36{,}000$, so $3.6 \times 10^{3}$ is the one that is not equal.

Divide the coefficients and subtract the exponents, keeping the signs straight.

$\frac{6 \times 10^{-3}}{2 \times 10^{4}} = \frac{6}{2} \times 10^{-3-4} = 3 \times 10^{-7}$

Subtracting $4$ from $-3$ gives $-7$, and the coefficient $3$ is already in range.

The powers of ten match, so add the coefficients and keep the shared power: $9 + 1 = 10$, giving $10 \times 10^{5}$. The coefficient $10$ is not less than $10$, so renormalize.

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

Adding can also push the coefficient out of range, so renormalize just as you would after multiplying.

Each coefficient is between $1$ and $10$, so the number with the largest exponent is largest. Three numbers have exponent $5$ and one has exponent $6$.

$1.2 \times 10^{6} > 9.9 \times 10^{5}$

The exponent $6$ wins outright, even though its coefficient $1.2$ is the smallest. So $1.2 \times 10^{6}$ is largest.

Convert $4.5 \times 10^{-5} = 0.000045$, then test each option. $45 \times 10^{-6} = 45 \times 0.000001 = 0.000045$.

$4.5 \times 10^{-5} = 0.000045 = 45 \times 10^{-6}$

The distractors give $0.0045$, $0.00045$, and $0.0000045$, none of which equal $0.000045$. Only $45 \times 10^{-6}$ matches.

Divide the coefficients and subtract the exponents.

$\frac{8 \times 10^{5}}{4 \times 10^{5}} = \frac{8}{4} \times 10^{5-5} = 2 \times 10^{0}$

Equal exponents subtract to $0$, and $10^{0} = 1$, so the value is just $2$, written $2 \times 10^{0}$.

Divide the coefficients and subtract the exponents.

$\frac{5 \times 10^{8}}{2 \times 10^{3}} = \frac{5}{2} \times 10^{8-3} = 2.5 \times 10^{5}$

The coefficient $2.5$ is in range, so the answer is $2.5 \times 10^{5}$.

Convert the bounds: $5 \times 10^{3} = 5{,}000$ and $5 \times 10^{4} = 50{,}000$. Now check each candidate. $2 \times 10^{4} = 20{,}000$.

$5{,}000 < 20{,}000 < 50{,}000$

So $2 \times 10^{4}$ is between them. The others are $4{,}000$, $60{,}000$, and $700$, all outside the range.

Squaring multiplies the quantity by itself, so square the coefficient and apply the power rule $(10^{4})^{2} = 10^{4 \times 2}$ to the power of ten.

$(3 \times 10^{4})^{2} = 3^{2} \times 10^{4 \times 2} = 9 \times 10^{8}$

The coefficient $9$ is in range, so the answer is $9 \times 10^{8}$.