Properties of Powers are eight rules by which powers are multiplied, divided, raised to a power, and simplified. This page collects all formulas in one table: product and quotient of powers with the same base, power of a power, power of a product and quotient, zero, negative, and fractional exponent. Each property is explained with reasoning, followed by step-by-step examples, analysis of common sign errors, and an interactive trainer for immediate self-assessment.
What is a power: base and exponent
A power is a shorthand notation for multiplying the same factor multiple times. The notation \(a^n\) means that the number \(a\) is multiplied by itself \(n\) times:
Here, \(a\) is the base of the power (the number being multiplied), and \(n\) is the exponent (how many times we multiply). For example, \(3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81\): the base is 3, and the exponent is 4.
Two notations worth remembering immediately: \(a^1 = a\) (one 'copy' of the number) and \(a^2\) is 'squared', \(a^3\) is 'cubed'. All the rules below work with bases and exponents: some properties require identical bases, others require identical exponents. Distinguishing these two cases is more important than memorizing formulas by heart.
All Properties of Powers: Formula Table
It's convenient to have the eight basic properties of powers in front of you in a single table. The first five work for any exponent, and the last three extend the concept of powers to zero, negative, and fractional exponents. Each property is explained with reasoning, followed by step-by-step examples, analysis of common sign errors, and an interactive trainer for immediate self-assessment.
| Property | Formula |
|---|---|
| Product same bases — add exponents | am · an = am+n |
| Quotient same bases — subtract exponents | am : an = am−n |
| Power of a power multiply exponents | (am)n = amn |
| Power of a product raise each factor to the power | (ab)n = anbn |
| Power of a quotient raise numerator and denominator to the power | (a/b)n = an/bn |
| Zero exponent any non-zero number equals one | a0 = 1 |
| Negative exponent invert to a fraction | a−n = 1/an |
| Fractional exponent is the n-th root | am/n = ⁿ√(am) |
Multiplication and Division of Powers with the Same Base
These are the two most common operations with powers.
When multiplying, exponents are added; the base remains unchanged:
Why: the first factor has \(m\) identical numbers, the second has \(n\); in total, there are \(m+n\). For example, \(2^5 \cdot 2^3 = 2^{5+3} = 2^8 = 256\).
When dividing, exponents are subtracted (subtract the exponent of the divisor from the exponent of the dividend):
For example, \(11^9 : 11^6 = 11^{9-6} = 11^3 = 1331\).
Important: when adding and subtracting powers, the exponents are not added or subtracted. The expression \(2^3 + 2^4\) cannot be simplified into a single power: \(2^3 + 2^4 = 8 + 16 = 24\), whereas \(2^7 = 128\). These properties only apply to multiplication and division.
The base also remains unchanged during multiplication. \(2^3 \cdot 2^4\) is \(2^7\), not \(4^7\): we add the exponents, and keep the base as it is.
Power of a Power, Power of a Product, and Power of a Quotient
Power of a power — exponents are multiplied:
For example, \((4^3)^2 = 4^{3 \cdot 2} = 4^6 = 4096\).
Power of a product — each factor is raised to the power:
For example, \((2 \cdot 5)^3 = 2^3 \cdot 5^3 = 8 \cdot 125 = 1000\).
Power of a quotient — both the numerator and the denominator are raised to the power:
For example, \(\left(\frac{3}{4}\right)^2 = \frac{9}{16}\).
The last two formulas are often used from right to left — they allow multiplying powers with different bases but the same exponent: \(4^3 \cdot 25^3 = (4 \cdot 25)^3 = 100^3 = 1\,000\,000\).
If the bases and exponents are different, the expression is usually brought to a common base: it's observed that the numbers are powers of the same number. For example, \(9^4 = (3^2)^4 = 3^8\), and the product \(4^3 \cdot 8^2\) becomes \((2^2)^3 \cdot (2^3)^2 = 2^6 \cdot 2^6 = 2^{12}\).
Zero and Negative Exponent
Any non-zero number raised to the power of zero equals one:
This is not an invention but a consequence of the division rule: \(6^3 : 6^3 = 6^{3-3} = 6^0\), but the same quotient is \(216 : 216 = 1\). Therefore, \(6^0 = 1\). The expression \(0^0\) is undefined.
A negative exponent means one divided by the power:
For example, \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\), and \(10^{-2} = \frac{1}{100} = 0{,}01\).
A useful consequence: a fraction raised to a negative exponent is inverted, and the exponent becomes positive:
The minus sign in the exponent does not make the number negative. The notation \(2^{-3}\) is a positive fraction \(\frac{1}{8}\), not \(-8\). The sign of the exponent indicates 'inversion', not the sign of the result.
Most importantly: all eight properties from the table continue to work for zero and negative exponents as well — for example, \(x^{-2} \cdot x^{-5} = x^{-7}\).
Fractional Exponent and Roots
A fractional exponent is another way to write a root:
The denominator of the fraction indicates the degree of the root, and the numerator indicates the power of the radicand. Examples: \(16^{\frac{1}{2}} = \sqrt{16} = 4\), \(32^{\frac{1}{5}} = \sqrt[5]{32} = 2\), \(27^{\frac{2}{3}} = \left(\sqrt[3]{27}\right)^2 = 3^2 = 9\).
Restriction: for an even denominator \(n\), the base must be non-negative (\(a \geqslant 0\)) — roots of even degree of a negative number are not extracted in the school curriculum.
Why is this useful: the same properties apply to fractional exponents, making it convenient to multiply and divide roots by converting them to powers. For example, \(\sqrt{7} \cdot \sqrt[4]{7} = 7^{\frac{1}{2}} \cdot 7^{\frac{1}{4}} = 7^{\frac{3}{4}}\).
Power of a Negative Number: Where the Sign Gets Lost
If the base is negative, the sign of the result depends on the parity of the exponent:
- even exponent — the result is positive: \((-3)^2 = 9\);
- odd exponent — the result is negative: \((-3)^3 = -27\).
It is especially important to distinguish between notations with and without parentheses: \((-3)^2 = 9\), but \(-3^2 = -9\). In the second case, only the three is squared, and the minus sign remains in front. This is the most common sign error in tests.
It's also useful to remember powers of minus one: \((-1)^{100} = 1\), and \((-1)^{101} = -1\) — an even exponent gives \(+1\), an odd exponent gives \(-1\).
Order of Operations with Powers
When an expression contains both powers and standard operations, the order is as follows: first parentheses, then powers and roots, then multiplication and division, and finally addition and subtraction.
For example, \(5 + 2 \cdot 3^2 = 5 + 2 \cdot 9 = 5 + 18 = 23\): first, we square, and only then multiply and add.
Examples: How to Apply Properties of Powers
Example 1. Multiplication and Division in One Expression
Task. Simplify \(x^7 \cdot x^4 : x^9\).
Step 1. Multiplication: bases are the same, exponents are added — \(x^7 \cdot x^4 = x^{7+4} = x^{11}\).
Step 2. Division: exponents are subtracted — \(x^{11} : x^9 = x^{11-9} = x^2\).
Answer. \(x^2\) (for \(x \neq 0\)).
Example 2. Power of a Product with a Coefficient
Task. Raise \((3a^4)^3\) to the power.
Step 1. Each factor is raised to the power: \((3a^4)^3 = 3^3 \cdot (a^4)^3\).
Step 2. Calculate separately: \(3^3 = 27\), and for a power of a power, the exponents are multiplied — \((a^4)^3 = a^{4 \cdot 3} = a^{12}\).
Answer. \(27a^{12}\). A common mistake here is forgetting to cube the three and writing \(3a^{12}\).
Example 3. Negative Exponent in a Product
Task. Calculate \(8^{-2} \cdot 8^{5}\).
Step 1. Bases are the same — add the exponents according to the usual rule; the minus sign is simply part of the addition: \(8^{-2+5} = 8^{3}\).
Step 2. \(8^3 = 512\).
Answer. \(512\). Note: it was not necessary to expand \(8^{-2}\) into a fraction — the properties of powers work directly with negative exponents.
Example 4. Fraction to a Negative Exponent
Task. Calculate \(\left(\frac{2}{5}\right)^{-2}\).
Step 1. A negative exponent inverts the fraction: \(\left(\frac{2}{5}\right)^{-2} = \left(\frac{5}{2}\right)^{2}\).
Step 2. Square the numerator and the denominator: \(\left(\frac{5}{2} ight)^{2} = \frac{25}{4}\).
Answer. \(\frac{25}{4} = 6{,}25\).
Example 5. Fractional Exponent
Task. Calculate \(81^{\frac{3}{4}}\).
Step 1. The denominator of the exponent is the degree of the root: \(81^{\frac{3}{4}} = \left(\sqrt[4]{81}\right)^{3}\).
Step 2. \(\sqrt[4]{81} = 3\), because \(3^4 = 81\).
Step 3. Now, cube the result: \(3^3 = 27\).
Answer. \(27\). It's more convenient to extract the root first — the numbers will be smaller than if you calculate \(81^3\) first.
Example 6. Combined Expression
Task. Calculate \(\dfrac{(3^2)^3}{3^4}\).
Step 1. Power of a power in the numerator: \((3^2)^3 = 3^{2 \cdot 3} = 3^6\).
Step 2. Division of powers with the same base: \(3^6 : 3^4 = 3^{6-4} = 3^2\).
Step 3. \(3^2 = 9\).
Answer. \(9\). General rule: first simplify parentheses and powers of powers, then handle multiplication and division.
Common Mistakes with Power Operations
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Adding exponents when adding powers: writing $2^2 + 2^2 = 2^4$.
Exponents are only added during multiplication. Here, \(2^2 + 2^2 = 4 + 4 = 8\), while \(2^4 = 16\) — the equality is incorrect. The sum of powers cannot be simplified into a single power; it can only be factored out: \(2^2 + 2^2 = 2 \cdot 2^2 = 2^3\).
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Multiplying bases as well: $2^3 \cdot 2^4 = 4^7$.
The base remains the same when multiplying powers; only the exponent changes: \(2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128\).
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Assuming $(a+b)^n = a^n + b^n$, confusing sum with product.
You can only distribute a power to a product: \((2 \cdot 3)^2 = 2^2 \cdot 3^2 = 4 \cdot 9 = 36\) — correct. But \((2+3)^2 = 5^2 = 25\), whereas \(2^2 + 3^2 = 13\). For sums, you need the formulas for the square of a binomial, not the properties of powers.
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Assuming $a^0 = 0$.
On the contrary: any non-zero number raised to the power of zero equals one — \(9^0 = 1\). Only a base of zero (\(0^n = 0\) for \(n > 0\)) results in zero, and the expression \(0^0\) is undefined.
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Subtracting exponents in the wrong order: $3^7 : 3^2 = 3^{-5}$.
Subtract the exponent of the divisor from the exponent of the dividend, not the other way around: \(3^7 : 3^2 = 3^{7-2} = 3^5 = 243\). The order here is as important as in regular subtraction.
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Thinking that a negative exponent makes the number negative: $2^{-2} = -4$.
A negative exponent means a fraction, not a minus sign: \(2^{-2} = \frac{1}{2^2} = \frac{1}{4}\). The result remains positive.
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Losing the sign of a negative base: writing $(-4)^2 = -16$.
With an even exponent, the minus sign 'disappears': \((-4)^2 = (-4) \cdot (-4) = 16\). The notation without parentheses means something else: \(-4^2 = -(4^2) = -16\). Parentheses make all the difference.
Questions and Answers about Power Properties
Are exponents added or multiplied when multiplying powers?
When multiplying powers with the same base, the exponents are added, and the base remains unchanged: \(a^m \cdot a^n = a^{m+n}\). For example, \(6^2 \cdot 6^5 = 6^{7}\). Exponents are multiplied in another case — when raising a power to a power: \((a^m)^n = a^{mn}\).
Are exponents added or subtracted when dividing powers?
Subtracted: subtract the exponent of the divisor from the exponent of the dividend, \(a^m : a^n = a^{m-n}\) for \(a \neq 0\). For example, \(10^9 : 10^4 = 10^{5}\).
What does a number raised to the power of zero equal?
One, if the base is not zero: \(a^0 = 1\). This follows from the division rule — the quotient \(a^n : a^n\) is equal to both \(a^0\) and one simultaneously. The expression \(0^0\) is undefined in the school curriculum.
How to get rid of a negative exponent?
Invert the expression into a fraction: \(a^{-n} = \frac{1}{a^n}\). For example, \(x^{-5} = \frac{1}{x^5}\). If the base is itself a fraction, it is simply inverted, and the exponent becomes positive.
What are the base and exponent of a power?
In the notation \(a^n\), the number \(a\) is the base of the power, meaning the factor that is repeated; \(n\) is the exponent, which indicates how many such factors there are. In the notation \(5^4\), the base is 5, the exponent is 4, and the value itself is 625.
Can powers with different bases be multiplied?
You can combine them into a single power if the exponents are the same: \(a^n \cdot b^n = (ab)^n\). For example, \(2^4 \cdot 5^4 = 10^4 = 10\,000\). If both the bases and exponents are different, the expression is first brought to a common base (when possible); otherwise, it is left as is.
How many properties of powers are there in total?
There are eight main properties: product and quotient of powers with the same base, power of a power, power of a product, power of a quotient, zero exponent, negative exponent, and fractional exponent. Different textbooks group them differently, so you might see 'five properties' or 'ten properties', but the content is the same — the complete summary is in the table above.
What does a fractional exponent mean?
A root. The denominator of the fraction is the degree of the root, and the numerator is the power of the radicand: \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\). For example, \(16^{\frac{1}{2}} = \sqrt{16} = 4\). For an even denominator, the base must be non-negative.
In which grade are the properties of powers studied?
Powers with natural exponents and their basic properties are studied in 7th-grade algebra, powers with negative integer exponents in 8th grade, and powers with rational (fractional) exponents and roots in grades 9-11. These same properties are needed for OGE and ЕГЭ exams when simplifying expressions.
