Positive Integer Exponent
Definition of Positive Integer Exponent
When solving mathematical problems, you often have to multiply the same number several times. For example, if the side of a square is 5 cm, then its area is $5 \cdot 5 = 25 cm^2$; the product $5 \cdot 5 = 5^2$ reads as "five squared". Or, if the side of a cube is 5 cm, its volume is $5 \cdot 5 \cdot 5 = 125 cm^3$; the product $5 \cdot 5 \cdot 5 = 5^3$ reads as "five cubed".
When solving statistical or combinatorial problems (see Chapters 9 and 10 of this Guide), the number of the same factors can grow greatly. In this case, a similar notation is introduced for the product:
$$\underbrace{7\cdot 7\cdot 7\cdot 7\cdot 7}_{5\text{times}} = 7^5, \underbrace{\frac{2}{3}\cdot \frac{2}{3}\cdot \cdots \cdot \frac{2}{3}}_{11\text{times}} = \left(\frac{2}{3}\right)^{11}$$
Exponentiation of a number a with a positive natural exponent n>1 is repeated multiplication of n factors, each equal to a: $$ a^n = \underbrace{a\cdot a \cdot \cdots \cdot a}_{n\text{times}} $$ Exponentiation of a with n=1 forms the base case: $$ a^1 = a $$ In the notation $a^n$, the number а is called the base , n is – the exponent (or, the power).
$a^n$ can be read as
- a raised to the nth power,
- the nth power of a,
- a to the nth power,
- a to the nth.
Sign of Exponentiation Result
$0^n = 0$
Zero raised to any power equals zero
$a \gt 0 \Rightarrow a^n \gt 0$
A positive number raised to any power is positive
$ a \lt 0 \Rightarrow \left[ \begin{array}{cc} a^n \lt 0, n-odd \\ a^n \gt 0, n-even \end{array} \right.$
A negative number raised to an odd power is negative
A negative number raised to an even power is positive.
An important practical implication: $a^2 \ge 0$ , $∀a \in \Bbb R$ - the square of a real number is non-negative.
Order of Operations in Expressions with Exponents
Priority of Operations (no Brackets)
- Exponentiation
- Multiplication and division
- Addition and subtraction
For example: $2^3:12+15\cdot3^2-24$
Step 1. Perform exponentiation: 8:12+15$\cdot$9-24
Step 2. Perform multiplication and division: $\frac{2}{3}$+135-24
Step 3. Perform addition and subtraction: 111 $\frac{2}{3}$
Examples of Problem Solving
Example 1. Perform exponentiation:
a) $3^5 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 243 $
b) $ \left(-\frac{1}{6}\right)^{3} = \left(-\frac{1}{6}\right) \cdot \left(-\frac{1}{6}\right) \cdot \left(-\frac{1}{6}\right) = \left(-\frac{1}{216}\right) $
c) $ (-0,1)^4 = (-0,1) \cdot (-0,1) \cdot (-0,1) \cdot (-0,1) = 0,0001 $
d) $ (1 \frac{1}{3})^3 = (\frac{4}{3})^3 = \frac{4}{3} \cdot \frac{4}{3} \cdot \frac{4}{3} = \frac{64}{27} = 2 \frac{10}{27} $
Example 2. Write the number as a cube:
a) $64 = 4 \cdot 4 \cdot 4 = 4^3 $
b)$ 0,008 = 0,2 \cdot 0,2 \cdot 0,2 = 0,2^3$
c) $ 4 \frac{17}{27} = \frac{125}{27} = \frac{5}{3} \cdot \frac{5}{3} \cdot \frac{5}{3} = \left(\frac{5}{3}\right)^3 = \left( 1\frac{2}{3}\right)^3$
d)$ -729 = (-9)\cdot(-9)\cdot(-9) = (-9)^3$
Example 3. Compare two numbers:
a) $ (-4)^3 and 0$
$ \left. \begin{array}{l} -4 \lt 0 \\ 3- odd \quad power \end{array} \right\} \Rightarrow (-4)^3 \lt 0 $
b) $(-6,1)^3 and (-6,1)^2$
$ \left. \begin{array}{l} (-6,1)^3 \lt 0 \\ (-6,1)^2 \gt 0 \end{array} \right\} \Rightarrow (-6,1)^3 \lt 0 \lt (-6,1)^2 \Rightarrow (-6,1)^3 \lt (-6,1)^2 $
c) $ \left(\frac{2}{3}\right)^3 and \left(\frac{2}{3}\right)^4 $
$ \left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}, \left(\frac{2}{3}\right) = \frac{2^4}{3^4} = \frac{16}{81} $
$ \frac{8}{27} = \frac{24}{81} \gt \frac{16}{81} \Rightarrow \frac{8}{27} \gt \frac{16}{81} \Rightarrow \left(\frac{2}{3}\right)^3 \gt \left(\frac{2}{3}\right)^4$
d) $-\left(\frac{5}{7}\right)^{10} and \left(-\frac{5}{7}\right)^{10} $
$ \left. \begin{array}{l} -\left(\frac{5}{7}\right)^{10} \lt 0 \\ \left(-\frac{5}{7}\right)^{10} \gt 0 \end{array} \right\} \Rightarrow -\left(\frac{5}{7}\right)^{10} \lt \left(-\frac{5}{7}\right)^{10} $
Example 4. Evaluate the expression:
a) $-4^3+(-3)^3 = -(4\cdot4\cdot4)+(-3)\cdot(-3)\cdot(-3) = -64+(-27) = -91$
b) $ 5-4\cdot2^3 = 5-4\cdot8 = 5-32 = -27 $
c) $ 0,3\cdot2^5-0,4\cdot(-0,1)^3 = 0,3\cdot32-0,4\cdot(-0,001) = 9,6+0,0004 = 9,6004 $
d) $ 3^3-81\cdot\left(\frac{2}{3}\right)^3 = 27-81 \cdot \frac{8}{27} = 27-3\cdot8 = 27-24 = 3 $
Example 5. Evaluate the expressions and arrange them in ascending order:
$$ \frac{5^2+3^2}{2}, \left(\frac{5+3}{2}\right)^2, \left(\frac{5}{2}\right)^2 + \left(\frac{3}{2}\right)^2 $$
$ \frac{5^2+3^2}{2} = \frac{25+9}{2} = \frac{34}{2} =17 $
$ \left(\frac{5+3}{2}\right)^2 = \left(\frac{8}{2}\right)^2 = 4^2 = 16 $
$ \left(\frac{5}{2}\right)^2 + \left(\frac{3}{2}\right)^2 = \frac{25}{4}+\frac{9}{4} = \frac{34}{4} = 8,5 $
$$ \left(\frac{5}{2}\right)^2 + \left(\frac{3}{2}\right)^2 \lt \left(\frac{5+3}{2}\right)^2 \lt \frac{5^2+3^2}{2} $$