📐 Properties of Exponents

Sahabat Smaridasa — To make it easier to work with exponential forms, we must know the properties of exponents that will be used in solving problems. These properties of exponents are very important and play the most crucial role in exponentiation. Therefore, if you want to be proficient and comfortable in mastering and solving exponent problems, we must first master the properties well and correctly.

There are many properties of exponents or exponentiation that we must memorize. However, it is important to remember that memorizing alone is not enough; we must also know how to use each property properly. If we can remember and use all these properties well, then we can say we have successfully learned them. Don't worry, in this article we have prepared the properties along with examples for each.

Problems that are directly related to properties of exponents or exponentiation always appear every year, both in national exams and in university entrance tests. This means that by mastering the properties of exponents well, at least one problem is guaranteed to be solvable. To deepen the use of exponent properties, please read and work on the collection of exponent problems available.

🔢 Properties of Exponents Based on the Exponent

The properties of exponents can be classified based on the type of exponent: positive integers, zero, negative integers, and fractions.

(i) Positive integer exponents ( \(m, n \in \mathbb{Z}^+\) )

\[ \begin{aligned} 1)&\; a^{m} \cdot a^{n} = a^{m+n} \\ 2)&\; \frac{a^{m}}{a^{n}} = a^{m-n} \\ 3)&\; (a^{m})^{n} = a^{m \cdot n} \\ 4)&\; (ab)^{m} = a^{m} \cdot b^{m} \\ 5)&\; \left(\frac{a}{b}\right)^{m} = \frac{a^{m}}{b^{m}} \end{aligned} \]

(ii) Zero exponent

\[ a^{0} = 1, \quad \text{with the condition } a \neq 0 \]

(iii) Negative integer exponent ( \(n\) positive)

\[ a^{-n} = \frac{1}{a^{n}} \quad \text{or} \quad \frac{1}{a^{-n}} = a^{n} \]

(iv) Fractional exponents

\[ \begin{aligned} 1)&\; a^{\frac{1}{n}} = \sqrt[n]{a} \\ 2)&\; a^{\frac{m}{n}} = \sqrt[n]{a^{m}} = \left(\sqrt[n]{a}\right)^{m} \end{aligned} \]

Below are some examples for the properties of exponents mentioned above.

Example 1

Determine the value of \(2^{3} \cdot 2^{4}\) and \(5^{2} \cdot 5^{2}\).

Solution: \[ \begin{aligned} &2^{3} \cdot 2^{4} = 2^{3+4} = 2^{7} = 128 \\ &5^{2} \cdot 5^{2} = 5^{2+2} = 5^{4} = 625 \end{aligned} \]

Example 2

Determine the value of \(\dfrac{5^{4}}{5^{3}}\) and \(\dfrac{7^{8}}{7^{6}}\).

Solution: \[ \begin{aligned} &\frac{5^{4}}{5^{3}} = 5^{4-3} = 5^{1} = 5 \\ &\frac{7^{8}}{7^{6}} = 7^{8-6} = 7^{2} = 49 \end{aligned} \]

Example 3

Determine the value of \((3^{2})^{2}\).

Solution: \[ (3^{2})^{2} = 3^{2 \cdot 2} = 3^{4} = 81 \]

Example 4

Determine the value of \((2 \cdot 3)^{2}\) and \(5^{3} \cdot 2^{3}\).

Solution: \[ \begin{aligned} &(2 \cdot 3)^{2} = 2^{2} \cdot 3^{2} = 4 \cdot 9 = 36 \\ &5^{3} \cdot 2^{3} = (5 \cdot 2)^{3} = 10^{3} = 1000 \end{aligned} \]

Example 5

Determine the value of \(\left(\dfrac{2}{5}\right)^{2}\) and \(\dfrac{6^{3}}{2^{3}}\).

Solution: \[ \begin{aligned} &\left(\frac{2}{5}\right)^{2} = \frac{2^{2}}{5^{2}} = \frac{4}{25} \\ &\frac{6^{3}}{2^{3}} = \left(\frac{6}{2}\right)^{3} = 3^{3} = 27 \end{aligned} \]

Example 6

Determine the value of \(5^{0}\) and \(\left(-\frac{1}{3}\right)^{0}\).

Solution: \[ \begin{aligned} &5^{0} = 1 \\ &\left(-\frac{1}{3}\right)^{0} = 1 \end{aligned} \]

Example 7

Determine the value of \(2^{-3}\), \(\dfrac{1}{2^{-2}}\), and \(\dfrac{3}{2^{-4}}\).

Solution: \[ \begin{aligned} &2^{-3} = \frac{1}{2^{3}} = \frac{1}{8} \\ &\frac{1}{2^{-2}} = 2^{2} = 4 \\ &\frac{3}{2^{-4}} = 3 \cdot 2^{4} = 3 \cdot 16 = 48 \end{aligned} \]

Example 8

Determine the value of \(4^{\frac{1}{2}}\) and \(3^{\frac{1}{5}}\).

Solution: \[ \begin{aligned} &4^{\frac{1}{2}} = \sqrt{4} = 2 \\ &3^{\frac{1}{5}} = \sqrt[5]{3} \end{aligned} \]

Example 9

Determine the value of \(3^{\frac{2}{3}}\) and \(16^{\frac{3}{4}}\).

Solution: \[ \begin{aligned} &3^{\frac{2}{3}} = \sqrt[3]{3^{2}} = \sqrt[3]{9} \\ &16^{\frac{3}{4}} = \left(\sqrt[4]{16}\right)^{3} = 2^{3} = 8 \\ &\text{Alternative: } 16^{\frac{3}{4}} = (2^{4})^{\frac{3}{4}} = 2^{3} = 8 \end{aligned} \]

Note: For \(n = 2\), the number 2 does not need to be written in the fractional exponent, as it is commonly known as the square root form.
\[ \begin{aligned} &2^{\frac{1}{2}} \text{ can be written as } \sqrt{2} \\ &\text{Method: } 2^{\frac{1}{2}} = \sqrt[2]{2} = \sqrt{2} \\ &3^{\frac{5}{2}} \text{ can be written as } \sqrt{3^{5}} \\ &\text{Method: } 3^{\frac{5}{2}} = \sqrt[2]{3^{5}} = \sqrt{3^{5}} \end{aligned} \]

⚠️ Exponent properties with negative base

\[ (-a)^{n} = \begin{cases} a^{n}, & \text{for } n \text{ even} \\ -(a^{n}), & \text{for } n \text{ odd} \end{cases} \]

Example 10

Determine the value of \((-2)^{4}\) and \((-2)^{5}\).

Solution: \[ \begin{aligned} &(-2)^{4} = 2^{4} = 16 \quad (\text{even exponent, positive result}) \\ &(-2)^{5} = -(2^{5}) = -32 \quad (\text{odd exponent, negative result}) \end{aligned} \]

We hope the examples above help us understand the properties of exponents, which I think are quite numerous. The best way to remember all properties of exponents is to continuously practice with exponent problems. We are confident that with practice, you will naturally remember and master them, including how to use each property.

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💡 Tip: The best way to master all properties is through continuous practice. Work through many exponent problems until you naturally remember each rule and its application.