Standard form is a number between 1 and 10, times a power of 10.

The digits give \(4.9\), and the decimal point has to move 5 places to the right to get back to 490,000.

\(490,000 = 4.9 \times 10^{ 5 }\)

A negative power of 10 moves the decimal point to the left, once for each power.

Moving the point in \(1.8\) 2 places to the left gives \(0.018\)

The number between 1 and 10 is \(7.8\).

Getting from \(7.8\) back to \(0.0078\) means moving the point 3 places to the left, so the power is \(-3\).

\(0.0078 = 7.8 \times 10^{ -3 }\)

A number in standard form is \(a \times 10^n\) with \(1 \leq a < 10\) and \(n\) a whole number. Test each number against both rules.

A: \(0.1 \times 10^{ -4 }\), the number in front is less than 1.

B: \(7.1 \times 10^{ 0.5 }\), the power of 10 is not a whole number.

C: \(53.8 \times 10^{ 7 }\), the number in front is 10 or more.

D: \(6 \times 10^{ 6 }\), the number in front is between 1 and 10 and the power is a whole number.

Only D meets both rules, so the answer is D.

Compare the powers of 10 first: the higher the power, the bigger the number, whatever the digits in front say.

Two of them share the power \(-3\), so those two are separated by the numbers in front: \(2.7 < 4.8\).

In order, smallest first:

\(2.7 \times 10^{ -3 }, \; 4.8 \times 10^{ -3 }, \; 2.1 \times 10^{ -1 }, \; 3.3 \times 10^{ 2 }\)

In standard form the number in front is between 1 and 10, and 59 is not.

Dividing it by 10 once gives 5.9. Every move of the point there has to be paid for the other way round in the power of 10, so the power goes up by 1.

\(59 \times 10^{ 7 } = 5.9 \times 10^{ 8 }\)

The number in front has to be between 1 and 10, and 0.06 is not.

Multiplying it by 10 twice gives 6. Every move of the point there is paid for the other way round in the power of 10, so the power goes down by 2.

\(0.06 \times 10^{ -3 } = 6 \times 10^{ -5 }\)

Write each number as a number between 1 and 10 times a power of 10: 980 \(= 9.8 \times 10^{ 2 }\) and 8 \(= 8 \times 10^{ 0 }\)

Divide the numbers in front and subtract the powers of 10.

\(9.8 ÷ 8 = 1.225\) and \(10^{ 2 } ÷ 10^{ 0 } = 10^{ 2 }\)

The number in front, \(1.225\), is already between 1 and 10, so the answer is:

\(1.225 \times 10^{ 2 }\)

Write each number as a single digit times a power of 10, counting the places the decimal point moves: 0.05 \(= 5 \times 10^{ -2 }\) and 4 \(= 4 \times 10^{ 0 }\)

Multiply the numbers in front and add the powers of 10.

\(5 × 4 = 20\) and \(10^{ -2 } × 10^{ 0 } = 10^{ -2 }\)

The number in front, \(20\), is 10 or more, so divide it by 10 and add 1 to the power:

\(2 \times 10^{ -1 }\)

Multiply the numbers in front, and add the powers of 10.

\(6 \times 1.2 = 7.2\) and \(10^{ 3 } \times 10^{ -2 } = 10^{ 1 }\)

That gives \(7.2 \times 10^{ 1 }\).

That number is already between 1 and 10, so this is standard form.

\(7.2 \times 10^{ 1 }\)

Divide the numbers in front, and subtract the powers of 10.

\(3 \div 2 = 1.5\) and \(10^{ 5 } \div 10^{ 2 } = 10^{ 3 }\)

That gives \(1.5 \times 10^{ 3 }\).

That number is already between 1 and 10, so this is standard form.

\(1.5 \times 10^{ 3 }\)

The height is the thickness of one sheet multiplied by the number of sheets.

Write the number of sheets in standard form too: \(7,000 = 7 \times 10^{3}\)

Multiply the numbers in front, and add the powers of 10.

\(4.6 \times 7 = 32.2\) and \(10^{-5} \times 10^{3} = 10^{-2}\)

That gives \(32.2 \times 10^{-2}\).

That number is not between 1 and 10, so divide it by 10 and add 1 to the power.

Height \(= 3.22 \times 10^{ -1 }\) metres

Asking how many times one quantity goes into another is asking for a division, so divide the bigger quantity by the smaller one.

Divide the numbers in front, and subtract the powers of 10.

\(4.62 \div 7 = 0.66\) and \(10^{ 6 } \div 10^{ 3 } = 10^{ 3 }\)

\(0.66 \times 10^{ 3 } = 660\)

There are 1000 grams in 1 kilogram, so you divide by 1000.

\(120 \div 1000 = 0.12\)

Now write 0.12 in standard form. The number between 1 and 10 is 1.2, and getting from 1.2 to 0.12 means moving the decimal point 1 place to the left.

\(0.12 = 1.2 \times 10^{ -1 }\)

Write every number as an ordinary number so they can be compared.

A is already ordinary: \(619\).

B: \(4.1 \times 10^{ 2 } = 410\).

C: \(53 \times 10^{ 1 } = 530\). C is not in standard form, so its power of ten, \(1\), does not say how big it is. Written in standard form it is \(5.3 \times 10^{ 2 }\), the same power of ten as B, so the numbers in front decide which is bigger.

D: \(6.8^{ 4 } = 2138.1376\), found with a calculator. The index 4 means \(6.8\) is a factor 4 times over, which is not the same as \(6.8 \times 4\).

In order, from smallest to largest:

\(410, \; 530, \; 619, \; 2138.1376\)

So the order is B C A D.