Magnitude

The power of ten for the number 1,000 is $10^{3}$, as $10 \times 10 \times 10 = 1,000$.

To write 0.005 in scientific notation, you would express it as $5 \times 10^{-3}$.

$10^{5}$ has a greater magnitude than $10^{-2}$ because it has a positive index, indicating a much larger number.

The basketball player's height is $2 \times 10^{0}$ metres (since $2$ is already between 1 and 10, the index is $0$), and the diameter of the basketball is $2.4 \times 10^{-1}$ metres.

The skyscraper's height is $3 \times 10^{2}$ metres, and the average human's height is $1.7 \times 10^{0}$ metres. The skyscraper uses the index $2$ because it is in the hundreds, whereas the human height uses the index $0$ as it is between 1 and 10.

The elephant's weight is $6 \times 10^{3}$ kilograms, and the house cat's weight is $5 \times 10^{0}$ kilograms. The elephant's weight uses the index $3$ as it is in the thousands, while the cat's weight uses the index $0$ because it is a single-digit number.

The height of the redwood tree in scientific notation is $1.155 \times 10^{2}$ metres, and David's height is $1.67 \times 10^{2}$ centimetres. To compare their heights in metres, we need to convert David's height into metres: $1.67 \times 10^{0}$ metres. Then, $10^{2}$ (redwood tree) ÷ $10^{0}$ (David) = $10^{2}$ times taller.

The distance to the Moon in scientific notation is $3.844 \times 10^{5}$ kilometres. A typical marathon distance is 42.195 kilometres, which can be expressed as $4.2195 \times 10^{1}$ kilometres. Comparing these using powers of ten, we find that $10^{5}$ (distance to the Moon)$ \div 10^{1}$ (marathon) = $10^{4}$ times longer.

The volume of the Pacific Ocean in scientific notation is $7.1 \times 10^{8}$ cubic kilometres. An Olympic-size swimming pool holds about 2,500 cubic metres of water, which is $2.5 \times 10^{3}$ cubic metres. Converting this to cubic kilometres since $1km^{3} = 10^{9} m^{3}$, the pool's volume is $2.5 \times 10^{-6}$ cubic kilometres. The ocean's volume is therefore $10^{8} \div 10^{-6} = 10^{14}$ times greater than that of the pool.

In mathematics, magnitude refers to the size or extent of a number, typically represented in terms of its power or scale, often using powers of ten.

You would write 0.00056 in powers of ten as $5.6 \times 10^{-4}$ in scientific notation.

Powers of ten are used because they provide a clear and consistent way to scale numbers up or down, making it easier to compare very large or very small quantities.

Scientific notation is a method of writing numbers as the product of a single-digit number and a power of ten, making it easier to work with and compare extremely large or small values.

Yes, powers of ten can help estimate distances, sizes, and quantities in everyday life, simplifying complex calculations and comparisons.

To compare magnitudes, you look at the indices in the scientific notation; the higher the index, the greater the magnitude of the number.

The power of ten for one million is $10^{6}$, as $10$ raised to the power of $6$ equals one million.

The metric system is based on powers of ten, making it easy to convert between units by simply moving the decimal point.

While not every number is a power of ten, every number can be expressed in a form that involves a power of ten, thanks to scientific notation.

No, magnitude can also be used to compare other aspects like brightness in astronomy or intensity in earthquakes, not just physical sizes.