Real numbers and scientific notation

$\sqrt{5} \approx 2.24$ , $\dfrac{7}{3} \approx 2.33$ , $\pi \approx 3.14$ . Largest is $\pi$ .
$\sqrt{2} + \sqrt{2} = 2\sqrt{2}$ . If this were rational, then $\sqrt{2} = \dfrac{r}{2}$ for some rational $r$ , making $\sqrt{2}$ rational — contradiction. So $2\sqrt{2}$ is irrational.
$\dfrac{5.97 \times 10^{24}}{7.35 \times 10^{22}} = \dfrac{5.97}{7.35} \times 10^2 \approx 0.8122 \times 10^2 = 81.2$ . The Earth is approximately $81$ times heavier.
$7 \times 10^{-5} \text{ m} = 70 \times 10^{-6} \text{ m} = 70\;\mu\text{m}$ .
$7.2 \times 4 = 28.8$ ; $10^{-3} \times 10^5 = 10^2$ . So $28.8 \times 10^2 = 2.88 \times 10^3$ .
Suppose $r$ is rational and $x$ is irrational and $r + x = q$ is rational. Then $x = q - r$ , a difference of two rationals, which is rational — contradicting $x$ being irrational.
$4.2^2 = 17.64$ and $4.3^2 = 18.49$ . Since $17.64 < 18 < 18.49$ , we have $4.2 < \sqrt{18} < 4.3$ .
$t = \dfrac{4.5 \times 10^{12}}{3 \times 10^8} = 1.5 \times 10^4 = 15\,000$ seconds (about $4.2$ hours).

Rational. $\sqrt{2} \times \sqrt{8} = \sqrt{16} = 4$ .
$4.2 \times 10^{-7} \text{ m} = 420 \times 10^{-9} \text{ m} = 420$ nm.
$\sqrt{2} \times \sqrt{2} = 2$ , which is rational. This works because the definition says each individual number is irrational, not that products of irrationals must be irrational.
$\dfrac{9 \times 10^{11}}{2.6 \times 10^7} = \dfrac{9}{2.6} \times 10^4 \approx 3.46 \times 10^4 = 34\,615$ dollars per person.
Let $x = 0.\overline{9}$ . Then $10x = 9.\overline{9}$ , so $10x - x = 9$ , giving $9x = 9$ and $x = 1$ . This shows $0.\overline{9}$ and $1$ are the same number — every terminating decimal also has a repeating representation. It does not blur the rational/irrational boundary; both forms are rational.

Assume $r + x = q$ where $q$ is rational. Then $x = q - r$ , a difference of two rationals, which is rational. This contradicts $x$ being irrational, so $r + x$ must be irrational.
$V = \dfrac{4}{3}\pi (4.4 \times 10^{26})^3 = \dfrac{4}{3}\pi \times 8.5184 \times 10^{79} \approx 4.19 \times 8.5184 \times 10^{79} \approx 3.6 \times 10^{80}$ m $^3$ .
Numerator: $(2.4)^2 \times (10^5)^2 = 5.76 \times 10^{10}$ . Division: $\dfrac{5.76 \times 10^{10}}{6 \times 10^3} = 0.96 \times 10^7 = 9.6 \times 10^6$ .
Operations per year: $3.6 \times 10^{12} \times 3.15 \times 10^7 = 11.34 \times 10^{19} = 1.134 \times 10^{20}$ . Audio processed: $1.134 \times 10^{20} \times 8 \times 10^{-9} = 9.072 \times 10^{11}$ seconds $= \dfrac{9.072 \times 10^{11}}{3600} \approx 2.52 \times 10^8$ hours.

Tier 1