0580 · 1.8Standard form

Powers and standard form rules

0:00A distance of several million metres and a measurement smaller than a thousandth of a metre look very different on paper. Both can involve strings of zeros that are easy to misread. Standard form gives us a compact way to write them. By the end, you will understand how it works, how to convert numbers, and how to calculate with them. The idea is to separate a number into its important digits and its scale.

0:22In four million five hundred thousand, the digits four and five describe the amount, while the remaining places establish its size. We can write that number as 4.5 × 1 million. Standard form writes the million as a power of ten. We need a brief reminder about powers of ten. 10² is one hundred, and 10³ is one thousand. A negative exponent means a reciprocal, so 10⁻³ is one thousandth.

0:47Multiplying by that value makes a number smaller rather than making it negative. Each time the exponent decreases by one, the value is divided by ten. This pattern continues through exponent zero, where the value is one, and into negative exponents. That explains why negative powers represent small positive values. The exponent tells us the scale, not whether the number itself is positive or negative. Standard form is written as A multiplied by 10ⁿ.

1:14The number A must be greater than or equal to one and less than ten. The exponent n is an integer, with positive and negative values used for large and small numbers. A is called the coefficient, and it contains exactly one nonzero digit before the decimal point. That restriction gives us one consistent way to write each positive number. 45 × 10⁵ has the correct value, but its coefficient is too large.

1:384.5 × 10⁶ is standard form. The coefficient becomes ten times smaller, so the power of ten must become ten times larger.

Converting standard form

1:47For our first worked example, write seventy-two thousand in standard form. Divide seventy-two thousand by ten thousand to obtain the coefficient seven point two. Since 10000 = 10⁴, write 7.2 × 10⁴. Multiplying these parts reconstructs the original number, so the value has not changed. A quick method is to count how many place value positions separate the original number from the coefficient. Here, four divisions by ten turn seventy-two thousand into seven point two.

2:17The positive exponent records the multiplication needed to rebuild the original value. Count positions rather than zeros, because numbers without trailing zeros follow the same rule. For our second example, write the decimal zero point zero zero zero five six in standard form. Multiply the original number by ten thousand to obtain five point six. Therefore, the original value is 5.6 ÷ 10000. This gives 5.6 × 10⁻⁴. The negative exponent compensates for making the coefficient larger.

2:47Five point six is much larger than the original decimal, so we need a small multiplier to restore its value. This is more reliable than remembering an unexplained direction rule. For a positive number smaller than one, its standard form exponent will be negative. Converting out of standard form simply reverses the process. 3.08 × 10⁵ means multiplying three point zero eight by one hundred thousand. The result is three hundred and eight thousand.

3:136.4 × 10⁻³ means dividing six point four by one thousand. The result is zero point zero zero six four. Zeros still matter when they hold a place between significant digits. Writing three point eight instead of three point zero eight gives a different value. Also, a coefficient of ten is not allowed, because the upper boundary is strictly less than ten. Numbers already between one and ten can use exponent zero, since multiplying by one leaves them unchanged.

Multiplying and dividing standard form

3:40Multiplication works by grouping the coefficients together and the powers of ten together. Multiplying powers with the same base means adding their exponents. For example, 1000 × 100 is one hundred thousand. That is why 10³ × 10² = 10⁵. Calculate the product of 3 × 10⁴ and 4 × 10⁻². Multiplying the coefficients gives twelve. Adding four and negative two gives exponent two, producing 12 × 10². Rewrite the result as 1.2 × 10³, because twelve is outside the permitted coefficient range.

4:17The final adjustment preserves the value rather than changing the answer. Dividing twelve by ten gives one point two. Increasing the exponent from two to three makes the scale ten times larger. These changes cancel each other, leaving the result equal to one thousand two hundred. Next, divide 4.8 × 10⁻³ by 6 × 10². Dividing four point eight by six gives zero point eight. Subtracting two from negative three gives negative five, producing 0.8 × 10⁻⁵.

4:48Rewrite this as 8 × 10⁻⁶. This time, the coefficient was too small rather than too large. Multiplying zero point eight by ten gives eight. Decreasing the exponent by one makes the scale ten times smaller, preserving the value. When dividing, subtract the exponent of the divisor and keep track of its sign.

Adding, subtracting and exam answers

5:08Addition and subtraction need a different approach because we are combining amounts, not multiplying scales. Think of three thousands plus two thousands as five thousands. The units match, so we can combine their counts. Standard form works similarly, but first both terms must use the same power of ten. Calculate 3.2 × 10⁵, plus 4.5 × 10⁴. Rewrite the second term as 0.45 × 10⁵. Add three point two and zero point four five to obtain three point six five.

5:40The result is 3.65 × 10⁵. Subtraction follows the same principle of matching the scale first. Rewrite 2 × 10⁻⁴ as 0.2 × 10⁻³. Subtract zero point two from five to obtain four point eight. The answer is 4.8 × 10⁻³. A common mistake is to add the exponents when two numbers in standard form are summed. The exponent rule belongs to multiplication, not to addition. Another mistake is stopping with a coefficient such as twelve or zero point eight.

6:14Always check both the value of your result and whether its coefficient lies in the required range. These methods must work without a calculator on Paper two. On Paper four, a calculator can help, but you still need to interpret the displayed exponent correctly. Use the scientific notation entry key appropriate to your calculator when entering standard form. A display using the letter E indicates a power of ten, not an extra factor to multiply separately.

6:39An exam question asks you to calculate this quotient and give your answer in standard form to three significant figures. Divide eight point four by three point six to obtain two point three recurring. Subtract two from six to obtain exponent four. Round the coefficient only at the end, giving 2.33 × 10⁴. Keep exact answers when possible, and follow any accuracy instruction given in the question. Otherwise, answers that are not exact are normally given to three significant figures.

7:08Do not round intermediate results, because that can affect the final answer. Working out the coefficient and the exponent on separate lines also makes your method clear. Standard form separates a number into a coefficient and a power of ten. Converting changes their balance while preserving the original value. For multiplication and division, combine coefficients and use the appropriate exponent rule. For addition and subtraction, match the powers first, then check that your final coefficient is at least one and less than ten.