From Zero to Engineer

Powers: multiplying, dividing and negative exponents

Review exercises on rounding and repeating decimals, then the laws of exponents: adding and subtracting exponents, the zero exponent, negative exponents and raising a power to a power.

A power is shorthand for repeated multiplication: instead of 10 · 10 · 10 · 10 we write 10 to the fourth. This lesson collects the laws of exponents - adding and subtracting exponents, the zero exponent, negative exponents and raising a power to a power.

Review exercises

1) Round each of the following decimal numbers to three significant figures, and then to two decimal places

$12.455$ → rounded to 3 significant figures is $12.5$

$12.455$ → rounded to 2 decimal places is $12.46$

$0.01356$ → rounded to 3 significant figures is $0.0136$

$0.01356$ → rounded to 2 decimal places is $0.01$

$0.1005$ → rounded to 3 significant figures is $0.101$

$0.1005$ → rounded to 2 decimal places is $0.10$

$1344.555$ → rounded to 3 significant figures is $1340$

$1344.555$ → rounded to 2 decimal places is $1344.56$

2) Write each of the following numbers in its simplest form

12.110110110\ldots = 12{,}(110)

0.123123123\ldots = 0{,}(123)

-3.11111\ldots = -3{,}(1)

-9360.936093609360 = -9360{,}(9360)

3) Convert each of the following common fractions to decimal form, correct to three decimal places

\tfrac{3}{16} = 0.188

\tfrac{-5}{9} = -0.556

\tfrac{7}{6} = 1.167

\tfrac{-24}{11} = -2.182

4) Convert each of the following decimal numbers to common fractions in their simplest form

0.6 = \tfrac{6}{10} = \tfrac{3}{5}

1{,}(4) = 1\tfrac{4}{9}

\begin{aligned}1{,}(4) &= 1 + 0{,}(4) \\10 \cdot 0{,}(4) &= 4{,}(4) \\9 \cdot 0{,}(4) &= 4 \\0{,}(4) &= \tfrac{4}{9}\end{aligned}

1{,}(24) = 1\tfrac{8}{33}

\begin{aligned}1{,}(24) &= 1 + 0{,}(24) \\100 \cdot 0{,}(24) &= 24{,}(24) \\99 \cdot 0{,}(24) &= 24 \\0{,}(24) &= \tfrac{24}{99} = \tfrac{8}{33}\end{aligned}

-7.3 = -\tfrac{73}{10} = -7\tfrac{3}{10}

Exponents and powers

10 \cdot 10 \cdot 10 \cdot 10 = 10^{4}

The four is the exponent.

The ten is the base.

Multiplying powers and adding the exponents

2^{4} \cdot 2^{3} = 2^{4+3} = 2^{7}

3^{4} \cdot 5^{4} = (3 \cdot 5)^{4} = 15^{4}

8^{3} \cdot 8^{5} = 8^{3+5} = 8^{8}

2^{3} \cdot 4^{3} = (2 \cdot 4)^{3} = 8^{3}

Dividing powers and subtracting the exponents

5^{6} : 5^{2} = 5^{6-2} = 5^{4}

12^{7} : 12^{3} = 12^{7-3} = 12^{4}

The zero exponent

1 = 3^{1} : 3^{1} = 3^{1-1} = 3^{0}

Therefore, any number raised to the power of 0 is equal to 1.

Powers with negative exponents

A negative exponent takes the reciprocal of the base.

6^{-2} = \tfrac{1}{6^{2}} = \tfrac{1}{36}

3^{-5} = \tfrac{1}{3^{5}} = \tfrac{1}{243}

Raising a power to a power

(5^{2})^{3} = 5^{2 \cdot 3} = 5^{6}

(4^{2})^{4} = 4^{2 \cdot 4} = 4^{8}

(5^{2})^{3} \neq 5^{(2^{3})}

5^{6} \neq 5^{8}

Frequently asked questions

How do you multiply powers with the same base?

Add the exponents and keep the base unchanged. For example 2⁴ · 2³ = 2⁷. When dividing, subtract the exponents instead.

Why is any number to the power of zero equal to one?

Because dividing a power by itself gives one, while subtracting the exponents gives zero. So 3¹ : 3¹ is both 1 and 3⁰.

What does a negative exponent mean?

The reciprocal of the base raised to the positive exponent. So 6⁻² means 1 divided by 6², that is one thirty-sixth.

Is (5²)³ the same as 5^(2³)?

No. In the first case the exponents are multiplied, giving 5⁶; in the second, 2³ is computed first, giving 5⁸. These are two different numbers.

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Decimals, rounding and repeating decimals

Roots, fractional exponents and scientific notation