Enter the two numbers X and Y, each split into a coefficient (the digits) and an exponent (the power of 10), and choose the operation. The equation below is linked to the fields, so you can also type the coefficients or exponents right into it. The result is shown in scientific, E and decimal notation, with steps such as "match the exponents" and "add the exponents".
Enter each coefficient as a number such as 1.23 (it does not have to be between 1 and 10), and each exponent as a whole number from −99 to 99. For "Square root √X" and "Square X²", the Y fields are not used (anything in them is ignored).
Result
Enter the coefficients and exponents of X and Y in the fields on the left and press "Calculate". The result (in scientific, E and decimal notation) and the steps will appear here.
Table of Contents
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What you can do on this page
-
What is this calculation used for?
-
How to Use
-
Formula
-
Symbols and terms
-
Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Add, subtract, multiply, divide, raise to a power, take the square root of, or square numbers in scientific notation (the form \(a \times 10^{n}\)), such as \((1.23 \times 10^{7}) + (3.45 \times 10^{2})\)
- The steps show rules such as "match the exponents, then add the coefficients" and "multiply the coefficients, add the exponents", so you learn the method, not just the answer
- The result is shown in scientific, E and decimal notation. You can also set the significant digits (how many digits to round the coefficient to) from 1 to 30 (10 if left blank)
- Check calculations with very large or very small numbers without mistakes, such as "speed of light × time" in physics or mole calculations in chemistry
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
This page handles exponents (powers of 10) from −99 to 99 and coefficients with up to 30 significant figures. The exponent of the result must also be from −99 to 99. To convert between ordinary numbers and scientific notation, use the companion Scientific Notation Converter page.
What is this calculation used for?
How far light travels in a year (one light-year)
The speed of light is about \(3.0 \times 10^{8}\) m/s, and a year is about \(3.15 \times 10^{7}\) seconds. The distance light travels in a year is a multiplication: the coefficients give \(3.0 \times 3.15 = 9.45\) and the exponents give \(8 + 7 = 15\), so it is about \(9.5 \times 10^{15}\) m (one light-year is about \(9.46 \times 10^{15}\) m).
The law of exponents shrinks a multiplication with 15 zeros into a two-digit calculation. That is its power.
Mole calculations in chemistry (multiplying by Avogadro's number)
1 mole of a substance contains about \(6.02 \times 10^{23}\) particles (Avogadro's number). For example, 0.5 mole of water contains \(0.5 \times 6.02 \times 10^{23} = 3.01 \times 10^{23}\) water molecules.
Mole calculations in high school chemistry are a chain of multiplying and dividing by Avogadro's number, so the rule on this page, handling coefficients and exponents separately, leads directly to better grades.
How long sunlight takes to reach Earth (distance ÷ speed)
The distance from the Sun to Earth is about \(1.5 \times 10^{11}\) m. Dividing by the speed of light, \(3 \times 10^{8}\) m/s, the coefficients give \(1.5 \div 3 = 0.5\) and the exponents give \(11 - 8 = 3\), so it is \(0.5 \times 10^{3} = 5 \times 10^{2}\) seconds, about 8 minutes 20 seconds.
In astronomy, both distances and speeds are huge, so subtracting exponents comes up in every division.
Comparing sizes (how much bigger is a person than a virus?)
A virus is about \(1 \times 10^{-7}\) m across, and a person is about \(1.7 \times 10^{0}\) m tall. Dividing gives \(1.7 \div 1 = 1.7\) and the exponent \(0 - (-7) = 7\), so a person is about \(1.7 \times 10^{7}\) times (17 million times) the size of a virus.
Even division with negative exponents works mechanically with the one rule "subtract the exponents".
Calculating with measurements and significant figures (science)
In science labs and reports, you often multiply measurements with significant figures. For example, a lot \(2.5 \times 10^{2}\) ft long and \(4.0 \times 10^{1}\) ft wide has an area of \(2.5 \times 4.0 = 10\) with exponent \(2 + 1 = 3\), which is \(10 \times 10^{3} = 1.0 \times 10^{4}\) ft².
Scientific notation keeps track of how many digits of the answer can be trusted (significant figures) as you calculate, so it is the standard for calculations based on measurements.
Formula
Adding and subtracting (match the exponents, then add or subtract the coefficients)
Standard notation
(the usual math form)
\(X + Y\)
\(=\)
\((\)
\(a\)
\(+\)
\(b\)
\()\)
\(\times\)
\(10\)
\(n\)
In words
(symbols replaced with words)
⑤
\(X + Y\): sum
\(=\)
\((\)
①
\(a\): coefficient of \(X\) after matching exponents
\(+\)
②
\(b\): coefficient of \(Y\) after matching exponents
\()\)
\(\times\)
③
\(10\): base
④
\(n\): matched exponent
The formula in words
①
Make the exponents of \(X\) and \(Y\) the same. Then add the
\(a\): coefficient of \(X\) after matching exponents
②
and the
\(b\): coefficient of \(Y\) after matching exponents
, and multiply the sum by the
③
\(10\): base
④
raised to the
\(n\): matched exponent
(that is, \(10^n\))
⑤
to get the
\(X + Y\): sum
(subtraction works the same way, subtracting the coefficients)
Quick example
Adding \(1.23 \times 10^{7}\) and \(3.45 \times 10^{2}\) (matching to the larger exponent, 7, gives \(3.45 \times 10^{2} = 0.0000345 \times 10^{7}\)):
sum (1.2300345 × 10⁷)
\(=\)
\((\)
1.23
\(+\)
0.0000345
\()\)
\(\times\)
base 10
matched exponent (7)
\(1.23 \times 10^{7} + 3.45 \times 10^{2} = (1.23 + 0.0000345) \times 10^{7}\)
\(= 1.2300345 \times 10^{7} = 12300345\)
Key idea
When the exponents differ, the place values do not line up, so you cannot just add the coefficients. Just as you line up place values when adding on paper, the key is to match the exponents (the place values) first. You can match to either exponent, but matching to the larger one is usual.
After calculating, if the coefficient's absolute value is no longer at least 1 and less than 10, normalize it (for example, \(9.2 \times 10^{3} + 8.3 \times 10^{3} = 17.5 \times 10^{3} = 1.75 \times 10^{4}\)).
Multiplying and dividing (multiply or divide the coefficients, add or subtract the exponents)
Standard notation
(the usual math form)
\(X \times Y\)
\(=\)
\((\)
\(a\)
\(\times\)
\(b\)
\()\)
\(\times\)
\(10\)
\(m + n\)
\(X \div Y\)
\(=\)
\((\)
\(a\)
\(\div\)
\(b\)
\()\)
\(\times\)
\(10\)
\(m - n\)
In words
(symbols replaced with words)
④
\(X \times Y\): product
\(=\)
\((\)
①
\(a\): coefficient of \(X\)
\(\times\)
②
\(b\): coefficient of \(Y\)
\()\)
\(\times\)
\(10\): base
③
\(m + n\): sum of exponents
⑧
\(X \div Y\): quotient
\(=\)
\((\)
⑤
\(a\): coefficient of \(X\)
\(\div\)
⑥
\(b\): coefficient of \(Y\)
\()\)
\(\times\)
\(10\): base
⑦
\(m - n\): difference of exponents
The formula in words
①
To multiply, multiply the
\(a\): coefficient of \(X\)
②
by the
\(b\): coefficient of \(Y\)
, then multiply by 10 to the power of the
③
\(m + n\): sum of exponents
(\(10^{m+n}\), with the exponents added)
④
to get the
\(X \times Y\): product
⑤
To divide, divide the
\(a\): coefficient of \(X\)
⑥
by the
\(b\): coefficient of \(Y\)
, then multiply by 10 to the power of the
⑦
\(m - n\): difference of exponents
(\(10^{m-n}\), with the exponents subtracted)
⑧
to get the
\(X \div Y\): quotient
Quick example
Multiplying \(2 \times 10^{4}\) by \(3 \times 10^{5}\):
product (6 × 10⁹)
\(=\)
\((\)
2
\(\times\)
3
\()\)
\(\times\)
base 10
sum of exponents (4 + 5 = 9)
\((2 \times 10^{4}) \times (3 \times 10^{5}) = (2 \times 3) \times 10^{4+5} = 6 \times 10^{9}\)
\((8 \times 10^{6}) \div (2 \times 10^{2}) = (8 \div 2) \times 10^{6-2} = 4 \times 10^{4}\)
Key idea
You add the exponents when multiplying because of the law of exponents \(10^{m} \times 10^{n} = 10^{m+n}\). Multiplying "10 multiplied \(m\) times" by "10 multiplied \(n\) times" gives 10 multiplied \(m + n\) times in total. Division is the reverse: \(10^{m} \div 10^{n} = 10^{m-n}\), so you subtract the exponents.
If the product of the coefficients is 10 or more, normalize it (for example, \((4 \times 10^{3}) \times (5 \times 10^{4}) = 20 \times 10^{7} = 2 \times 10^{8}\)).
Powers and square roots (multiply or halve the exponent)
Standard notation
(the usual math form)
\(X^{k}\)
\(=\)
\(a\)
\(k\)
\(\times\)
\(10\)
\(m k\)
\(\sqrt{X}\)
\(=\)
\(\sqrt{a}\)
\(\times\)
\(10\)
\(m/2\)
In words
(symbols replaced with words)
④
\(X^k\): X to the power k
\(=\)
①
\(a\): coefficient of \(X\)
②
\(k\): power
\(\times\)
\(10\): base
③
\(m \times k\): product of exponents
⑦
\(\sqrt{X}\): square root of X
\(=\)
⑤
\(\sqrt{a}\): square root of the coefficient
\(\times\)
\(10\): base
⑥
\(m \div 2\): half the exponent
The formula in words
①
For a power (to the \(k\)th power), raise the
\(a\): coefficient of \(X\)
②
to the
\(k\): power
, then multiply by 10 to the power of the
③
\(m \times k\): product of exponents
(\(10^{mk}\), with the exponent \(m\) multiplied by \(k\))
④
to get
\(X^k\): X to the power k
⑤
For a square root, multiply the
\(\sqrt{a}\): square root of the coefficient
⑥
by 10 to the power of
\(m \div 2\): half the exponent
⑦
to get the
\(\sqrt{X}\): square root of X
(if the exponent \(m\) is odd, first multiply the coefficient by 10 and lower the exponent by 1 to make it even, then halve it)
Quick example
The square of \(2 \times 10^{3}\) and the square root of \(4 \times 10^{6}\):
X squared (4 × 10⁶)
\(=\)
coefficient (2)
power (2)
\(\times\)
base 10
product of exponents (3 × 2 = 6)
\((2 \times 10^{3})^{2} = 2^{2} \times 10^{3 \times 2} = 4 \times 10^{6}\)
\(\sqrt{4 \times 10^{6}} = \sqrt{4} \times 10^{6 \div 2} = 2 \times 10^{3}\)
Key idea
You multiply the exponents for a power because of the law of exponents \((10^{m})^{k} = 10^{mk}\). Multiplying \(10^{m}\) by itself \(k\) times gives 10 multiplied \(m \times k\) times in total. A square root is "the number that gives back the original when squared", so it does the reverse and halves the exponent.
When the exponent is odd, it cannot be halved as is, so multiply the coefficient by 10 to make the exponent even first (for example, \(\sqrt{4 \times 10^{7}} = \sqrt{40 \times 10^{6}} = \sqrt{40} \times 10^{3} \approx 6.32 \times 10^{3}\)).
The trick with scientific notation is to handle the coefficients and the exponents separately. To add or subtract, match the exponents, then add or subtract the coefficients. To multiply or divide, multiply or divide the coefficients and add or subtract the exponents. For a power, multiply the exponent by \(k\); for a square root, halve it. Finally, normalize so the coefficient's absolute value is at least 1 and less than 10.
Symbols and terms
Symbols
| \(X\), \(Y\) | X Y | The two numbers in the calculation. On this page, each is entered as "coefficient × 10 to the power of the exponent" (for example, \(X = 1.23 \times 10^{7}\)). |
| \(a\), \(b\) | a b | The coefficients (the digits) of \(X\) and \(Y\). In normalized form, the absolute value is at least 1 and less than 10. |
| \(m\), \(n\) | m n | The exponents of \(X\) and \(Y\). The number of times 10 is multiplied, which carries the size of the number. A negative exponent stands for a small number less than 1. |
| \(k\) | k | The power, which tells what power to raise \(X\) to. On this page, it is the value of \(Y\) when you choose "Power X^Y". |
| \(10^{m+n}\) | 10 to the power m plus n | The power of 10 that appears in the law of exponents, the result of \(10^{m} \times 10^{n}\) (for example, \(10^{4} \times 10^{5} = 10^{9}\)). |
| \(\sqrt{X}\) | square root of X | The square root of \(X\): the number, 0 or greater, that gives back \(X\) when squared (for example, \(\sqrt{4 \times 10^{6}} = 2 \times 10^{3}\)). |
Terms
| law of exponents (laws of exponents) | Rules that turn calculations with powers into adding, subtracting or multiplying exponents. They are the heart of this page. The main ones are \(10^{a} \times 10^{b} = 10^{a+b}\) (multiplying adds the exponents, for example \(10^{4} \times 10^{5} = 10^{9}\)), \(10^{a} \div 10^{b} = 10^{a-b}\) (dividing subtracts them) and \((10^{a})^{b} = 10^{ab}\) (a power multiplies them). "10 multiplied 4 times" times "10 multiplied 5 times" is 10 multiplied 9 times in total. Counting like this, you can work out the rules without memorizing them. |
| coefficient (significand) | The \(a\) in \(a \times 10^{n}\) (the digits). The companion Scientific Notation Converter page explains it in detail. |
| exponent | The small number at the upper right of a number. On this page, it tells how many times to multiply by 10 (or, if negative, how many times to divide by 10), which carries the size of the number. |
| normalize | To adjust a number so the coefficient's absolute value is at least 1 and less than 10. If a calculation gives something like \(17.5 \times 10^{3}\), rewrite it as \(1.75 \times 10^{4}\). |
| significant digits (significant figures) | How many digits of a number, counted from the first nonzero digit, can be trusted and used. On this page, the coefficient of the result is rounded to this many digits. |
| E notation | A way of writing the \(\times 10^{n}\) part with the letter e (or E), for example 6e9 for \(6 \times 10^{9}\). The result on this page also has an E notation row. The companion Scientific Notation Converter page explains it in detail. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
| Powers and exponents (Grade 6) |
|
| Scientific notation basics (Grade 8) |
|
| Laws of exponents (Grade 8) |
|
| Negative exponents (Grade 8) |
|
| Square roots (Grade 8) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table for adding and subtracting (match the exponents, then combine the coefficients)
| Coefficient of X | 1.23 |
| Exponent of X | 7 |
| Coefficient of Y | 3.45 |
| Exponent of Y | 2 |
| X + Y | =B1*10^B2+B3*10^B4 |
| X − Y | =B1*10^B2-B3*10^B4 |
Table for multiplying and dividing (coefficients multiply or divide, exponents add or subtract)
| Coefficient of X | 8 |
| Exponent of X | 6 |
| Coefficient of Y | 2 |
| Exponent of Y | 2 |
| X × Y | =(B1*10^B2)*(B3*10^B4) |
| X ÷ Y | =(B1*10^B2)/(B3*10^B4) |
Table for powers and square roots (multiply or halve the exponent)
| Coefficient of X | 4 |
| Exponent of X | 6 |
| Power k | 2 |
| X to the power k | =(B1*10^B2)^B3 |
| √X (square root) | =SQRT(B1*10^B2) |
In the first table, enter the coefficients and exponents of X and Y in B1 to B4. B5 shows the sum (12300345 with the example values) and B6 the difference (12299655).
In the second table, B5 is the product (1600000000 = 1.6×10⁹ with the example values) and B6 is the quotient (40000 = 4×10⁴).
In the third table, B4 is the power (1.6×10¹³ with the example values) and B5 is the square root (2000 = 2×10³).
To see a result in scientific notation (E notation) such as 1.23E+07, change the result cell's format to Scientific (Home → Number Format → Scientific).
In the second table, B5 is the product (1600000000 = 1.6×10⁹ with the example values) and B6 is the quotient (40000 = 4×10⁴).
In the third table, B4 is the power (1.6×10¹³ with the example values) and B5 is the square root (2000 = 2×10³).
To see a result in scientific notation (E notation) such as 1.23E+07, change the result cell's format to Scientific (Home → Number Format → Scientific).
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table for adding and subtracting (match the exponents, then combine the coefficients)
| Coefficient of X | 1.23 |
| Exponent of X | 7 |
| Coefficient of Y | 3.45 |
| Exponent of Y | 2 |
| X + Y | =B1*10^B2+B3*10^B4 |
| X − Y | =B1*10^B2-B3*10^B4 |
Table for multiplying and dividing (coefficients multiply or divide, exponents add or subtract)
| Coefficient of X | 8 |
| Exponent of X | 6 |
| Coefficient of Y | 2 |
| Exponent of Y | 2 |
| X × Y | =(B1*10^B2)*(B3*10^B4) |
| X ÷ Y | =(B1*10^B2)/(B3*10^B4) |
Table for powers and square roots (multiply or halve the exponent)
| Coefficient of X | 4 |
| Exponent of X | 6 |
| Power k | 2 |
| X to the power k | =(B1*10^B2)^B3 |
| √X (square root) | =SQRT(B1*10^B2) |
The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace B1 to B4 (B1 to B3 in the third table) with your own numbers.
You can switch a result to scientific notation (E notation) with Format → Number → Scientific.
You can switch a result to scientific notation (E notation) with Format → Number → Scientific.
How to calculate it in Python
import math
x = 1.23e7 # X = 1.23×10^7 (can be written as an E notation literal)
y = 3.45e2 # Y = 3.45×10^2
print(x + y) # addition → 12300345.0
print(f"{x * y:.4e}") # multiplication in E notation → 4.2435e+09
print(f"{x ** 2:.4e}") # square → 1.5129e+14
print(f"{math.sqrt(x):.4e}") # square root → 3.5071e+03
Runs with the standard library only. In Python you can write numbers directly as E notation literals such as 1.23e7, and show results in scientific notation style with an f-string format such as ".4e" (E notation with 4 decimal places in the coefficient). Floating-point numbers are accurate to about 15 to 16 significant digits, so for more digits use the decimal module from the standard library. Change the numbers at the top and run it.
How to write it in LaTeX and other math languages (copy and paste)
Adding and subtracting (match the exponents, then add or subtract the coefficients)
a × 10ⁿ + b × 10ⁿ = (a + b) × 10ⁿ
a \times 10^{n} + b \times 10^{n} = (a + b) \times 10^{n}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>a</mi><mo>×</mo><msup><mn>10</mn><mi>n</mi></msup>
<mo>+</mo>
<mi>b</mi><mo>×</mo><msup><mn>10</mn><mi>n</mi></msup>
<mo>=</mo>
<mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>)</mo>
<mo>×</mo><msup><mn>10</mn><mi>n</mi></msup>
</mrow>
</math>
a * 10^n + b * 10^n = (a + b) * 10^n
1.23*10^7 + 3.45*10^2
evalf(1.23*10^7 + 3.45*10^2);
R = 1.23e7 + 3.45e2;
a × 10^n + b × 10^n = (a + b) × 10^n
Multiplying and dividing (multiply or divide the coefficients, add or subtract the exponents)
(a × 10ᵐ) × (b × 10ⁿ) = ab × 10ᵐ⁺ⁿ, (a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10ᵐ⁻ⁿ
(a \times 10^{m}) \times (b \times 10^{n}) = ab \times 10^{m+n}, \quad (a \times 10^{m}) \div (b \times 10^{n}) = \dfrac{a}{b} \times 10^{m-n}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mo>(</mo><mi>a</mi><mo>×</mo><msup><mn>10</mn><mi>m</mi></msup><mo>)</mo>
<mo>×</mo>
<mo>(</mo><mi>b</mi><mo>×</mo><msup><mn>10</mn><mi>n</mi></msup><mo>)</mo>
<mo>=</mo>
<mi>a</mi><mi>b</mi><mo>×</mo>
<msup><mn>10</mn><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup>
</mrow>
</math>
(a * 10^m) * (b * 10^n) = a b * 10^(m+n), (a * 10^m) -: (b * 10^n) = (a -: b) * 10^(m-n)
(2*10^4) * (3*10^5)
evalf((2*10^4) * (3*10^5));
R1 = (2e4) * (3e5); R2 = (8e6) / (2e2);
(a × 10^m) × (b × 10^n) = ab × 10^(m+n), (a × 10^m) ÷ (b × 10^n) = (a ÷ b) × 10^(m−n)
Powers and square roots (multiply or halve the exponent)
(a × 10ᵐ)ᵏ = aᵏ × 10ᵐᵏ
(a \times 10^{m})^{k} = a^{k} \times 10^{mk}, \quad \sqrt{a \times 10^{m}} = \sqrt{a} \times 10^{m/2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msup>
<mrow><mo>(</mo><mi>a</mi><mo>×</mo><msup><mn>10</mn><mi>m</mi></msup><mo>)</mo></mrow>
<mi>k</mi>
</msup>
<mo>=</mo>
<msup><mi>a</mi><mi>k</mi></msup>
<mo>×</mo>
<msup><mn>10</mn><mrow><mi>m</mi><mi>k</mi></mrow></msup>
</mrow>
</math>
(a * 10^m)^k = a^k * 10^(m k), sqrt(a * 10^m) = sqrt(a) * 10^(m/2)
{(2*10^3)^2, Sqrt[4*10^6]}
evalf((2*10^3)^2); evalf(sqrt(4*10^6));
R1 = (2e3)^2; R2 = sqrt(4e6);
(a × 10^m)^k = a^k × 10^(mk), √(a × 10^m) = √a × 10^(m/2)
How to have ChatGPT do the calculation
You are a calculation assistant for scientific notation. Do the following calculations by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). Calculate the following four, and show each answer both (a) in scientific notation (a coefficient with absolute value at least 1 and less than 10 × a power of 10) and (b) as an ordinary number (decimal notation). 1. (1.23 × 10^7) + (3.45 × 10^2) 2. (2 × 10^4) × (3 × 10^5) 3. (2 × 10^3) squared 4. √(4 × 10^6) Also explain how the steps "multiply the coefficients, add the exponents" were used in the multiplication. Show the code you used and the numbers from the execution result.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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Related Features
Scientific Notation Converter (b×10ⁿ, E Notation, Engineering Notation)
The companion page. Converts between ordinary numbers and scientific, E and engineering notation
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Calculates a power from the base and the exponent. Handy for checking how big 10 to the nth power is
Nth Root Calculator (Square Root, Cube Root, nth Root)
For a closer look at square roots (√) themselves. Cube roots and nth roots are covered too
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