The degree of the root becomes the denominator of the exponent. The cube root of eight is eight to the power of one third, and the fourth root of eighty-one is eighty-one to the power of one quarter.

Q: Why does the square root of a negative number not exist?

In the set of real numbers every square is non-negative, so no real number squared gives minus four. With an odd-degree root the problem disappears - the fifth root of minus thirty-two is minus two.

Q: What is scientific notation?

Writing a number as a mantissa in the interval from one up to but not including ten, multiplied by ten raised to the appropriate power. For example 0.000485 is 4.85 times ten to the power of minus four.

Q: How do you add numbers written in scientific notation?

First bring both terms to the same power of ten, then add the mantissas alone, and finally return to a form whose mantissa is less than ten.

Roots as powers with fractional exponents, multiplying and dividing by powers of ten, the order of operations, and writing numbers in scientific notation along with operations on them.

An exponent does not have to be a whole number. When it is a fraction, the power becomes a root - and that single observation lets you handle roots with the same rules as powers. The second half of the lesson covers scientific notation: a way of writing very large and very small numbers so they fit on one line.

Fractional exponents and roots

Roots can be expressed as powers with fractional (rational) exponents.

8^{\tfrac{1}{3}} = \sqrt[3]{8} = 2

6^{\tfrac{1}{5}} = \sqrt[5]{6} = 1.431

4^{\tfrac{1}{2}} = \sqrt{4} = \pm 2

81^{\tfrac{1}{4}} = \sqrt[4]{81} = \pm 3

(-32)^{\tfrac{1}{5}} = \sqrt[5]{-32} = -2

16^{\tfrac{1}{7}} = \sqrt[7]{16} = 1.486

8^{\tfrac{1}{2}} = \sqrt{8} = 2.828

19^{\tfrac{1}{4}} = \sqrt[4]{19} = \pm 2.088

\sqrt{-4}

the square root of a negative number does not exist (in the set of real numbers)

Multiplying and dividing whole numbers by powers of 10

0.012045 \cdot 10^{4} = 120.45

four places to the right

13.5074 \cdot 10^{-3} = 0.0135074

three places to the left

144.032 : 10^{5} = 144.032 \cdot 10^{-5} = 0.00144032

five places to the left

0.012045 : 10^{-2} = 0.012045 \cdot 10^{2} = 1.2045

two places to the right

The order of operations

exponentiation → multiplication / division → addition / subtraction

5(3 \cdot 4^{2} : 6 - 7) = 5(3 \cdot 16 : 6 - 7) = 5(48 : 6 - 7) = 5(8 - 7) = 5

Scientific notation

A decimal number in the interval <1, 10), called the mantissa, multiplied by 10 raised to the appropriate power.

0.000485 = 4.85 \cdot 10^{-4}

423.8 = 4.238 \cdot 10^{2}

52\,674 = 5.2674 \cdot 10^{4}

0.00723 = 7.23 \cdot 10^{-3}

0.0582 = 5.82 \cdot 10^{-2}

1\,523\,800 = 1.523800 \cdot 10^{6}

Multiplying and dividing numbers in scientific notation

\begin{aligned}\text{a)}\quad 472.3 \cdot 0.000564 &= (4.723 \cdot 10^{2}) \cdot (5.64 \cdot 10^{-4}) \\&= 26.6377 \cdot 10^{-2} \\&= 2.66377 \cdot 10 \cdot 10^{-2} \\&= 2.6638 \cdot 10^{-1}\end{aligned}

correct to four decimal places

\begin{aligned}\text{b)}\quad 752\,000 : 0.862 &= (7.52 \cdot 10^{5}) : (8.62 \cdot 10^{-1}) \\&= 0.87238 \cdot 10^{5} : 10^{-1} \\&= 0.87239 \cdot 10^{6} \\&= 8.7239 \cdot 10^{-1} \cdot 10^{6} \\&= 8.7239 \cdot 10^{5}\end{aligned}

10^{5} \cdot 10^{1}

dividing by 10⁻¹ is multiplying by 10¹

correct to four decimal places

Adding and subtracting numbers in scientific notation

\begin{aligned}\text{a)}\quad 43.6 \cdot 10^{2} + 8.12 \cdot 10^{3} &= 4.36 \cdot 10^{3} + 8.12 \cdot 10^{3} \\&= (4.36 + 8.12) \cdot 10^{3} \\&= 12.48 \cdot 10^{3} \\&= 1.248 \cdot 10^{4}\end{aligned}

\begin{aligned}\text{b)}\quad 7.84 \cdot 10^{5} - 12.36 \cdot 10^{3} &= 7.84 \cdot 10^{5} - 0.1236 \cdot 10^{5} \\&= (7.84 - 0.1236) \cdot 10^{5} \\&= 7.7164 \cdot 10^{5}\end{aligned}

\begin{aligned}\text{c)}\quad 4.25 \cdot 10^{-3} + 1.74 \cdot 10^{-2} &= 0.425 \cdot 10^{-2} + 1.74 \cdot 10^{-2} \\&= 2.165 \cdot 10^{-2}\end{aligned}

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Frequently asked questions

How do you rewrite a root as a power?

The degree of the root becomes the denominator of the exponent. The cube root of eight is eight to the power of one third, and the fourth root of eighty-one is eighty-one to the power of one quarter.

Why does the square root of a negative number not exist?

In the set of real numbers every square is non-negative, so no real number squared gives minus four. With an odd-degree root the problem disappears - the fifth root of minus thirty-two is minus two.

What is scientific notation?

Writing a number as a mantissa in the interval from one up to but not including ten, multiplied by ten raised to the appropriate power. For example 0.000485 is 4.85 times ten to the power of minus four.

How do you add numbers written in scientific notation?

First bring both terms to the same power of ten, then add the mantissas alone, and finally return to a form whose mantissa is less than ten.

A lesson from the PhiBoard knowledge base: phiboard.com/en/knowledge-base/from-zero-to-engineer/roots-and-scientific-notation. On the lesson page you will also find games and practice exercises - free and without an account.

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Powers: multiplying, dividing and negative exponents

Engineering notation and accuracy of calculations