Power of a power calculator
Collapse a nested power into a single one: (2^3)^4 is 2 to the 12th, which is 4096, because the two exponents multiply. You see the combined exponent and the value, and negative or fractional exponents work when the result is a real number. Free in your browser.
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- Your data stays on your device
Run — free
Worked example
- Base
- 2
- Inner exponent
- 3
- Outer exponent
- 4
- Value
- 4,096
- Simplified exponent
- 12
- Expression
- (2^3)^4 = 2^12 = 4096
A power of a power is the nested exponent form (a to the m) to the n that algebra textbooks collapse by multiplying the two exponents: the identity is (a^m)^n equals a to the power of m times n. Students, tutors, and exam writers search for a power of a power calculator when they need the simplified exponent and the numeric value without expanding the tower by hand. This tool accepts a finite base together with two finite exponents — the inner exponent m and the outer exponent n — multiplies m by n, and returns both the combined exponent and the evaluated value of the base raised to that product. Two cubed, then raised to the fourth, is two to the twelfth, which equals 4096.
What the power-of-a-power identity actually computes
The justification is elementary. a^m means a multiplied by itself m times when m is a positive integer; raising that block to the n means writing the block n times, which is m times n factors of a. The same multiplication extends by continuity and by the usual algebraic definitions to zero, negative, and fractional exponents whenever the resulting real power exists. So (a^2)^3 is a^6, (a^{-3})^2 is a^{-6}, and (a^{1/2})^4 is a^2.
This capability is not the product rule, which adds exponents of a product of like powers, and it is not the quotient rule, which subtracts them. It is also not a right-associated tower a^(m^n): the parentheses sit around the inner power, so the outer exponent applies to an already-powered quantity. Two cubed, then raised to the fourth, is (2^3)^4 = 8^4 = 4096, which equals 2^12, not 2^(3^4) = 2^81.
People who search for a power of a power calculator want that one identity, the combined exponent, and the evaluated number, not a dashboard of every exponent law. Use the tool whenever the base and both exponents are known and you need a^(m·n) written and computed.
Where simplifying nested powers helps
Classrooms and tutoring platforms use a dedicated power of a power calculator so learners can check the product of two exponents and the evaluated power without mixing the identity with the product rule that adds exponents. Exam-prep generators and learning-management plugins need a deterministic oracle that marks (2^3)^4 as 2^12 equals 4096, accepts signed and fractional exponents when the real power is defined, and rejects non-finite garbage instead of silently returning Infinity that a buggy grader might treat as success.
Science and engineering notebooks collapse nested powers in SI prefixes, geometric growth, and polynomial degrees — writing (10^3)^4 as 10^12 is the same multiplication of exponents. Spreadsheet users who keep stacking POWER functions often want a single combined exponent they can paste back into a formula. Keep requests small, pin the rounded fields in tests, and you get a reproducible simplified exponent and value without standing up a computer-algebra library for one multiplication and one power.
What you can do with it
Classroom checks of (a^m)^n
Let students submit a base and both exponents and verify the combined exponent and the evaluated power against the textbook identity.
Exam and LMS auto-grading
Mark power-of-a-power items with a deterministic oracle that multiplies exponents and rejects non-finite input instead of returning Infinity.
Notebooks collapsing nested powers
Rewrite (10^3)^4 as 10^12, or any nested SI, growth, or polynomial power, without importing a computer-algebra library.
FAQ
How do I simplify a power of a power?
Example: 2, 3, and 4 yield 12 and 4096, because (2^3)^4 = 2^12 = 4096.
What is the difference between (a^m)^n and a^(m^n)?
The parentheses sit around the inner power, so the exponents multiply: (2^3)^4 = 2^12 = 4096. A right-associated tower 2^(3^4) is 2^81, a different and much larger number. This calculator applies only the power-of-a-power identity.
Does the calculator handle negative or fractional exponents?
Yes, whenever the resulting real power is a finite number. (a^{-3})^2 becomes a^{-6}, and (a^{1/2})^4 becomes a^2. A negative base with a non-integer simplified exponent is rejected when the real power is not finite.
What happens if an input is missing or not a finite number?
NaN, Infinity, empty strings, and non-numeric values are rejected and are not billed.
Is the power of a power rule the same as the product rule?
No. The product rule adds the exponents when you multiply two powers of the same base. A power of a power multiplies them instead, so (2^3)^4 becomes 2 to the 12th, not 2 to the 7th.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/math/power-of-a-power \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"base":2,"inner_exponent":3,"outer_exponent":4}'const res = await fetch("https://api.kit.forhosting.com/math/power-of-a-power", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"base": 2,
"inner_exponent": 3,
"outer_exponent": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/power-of-a-power",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"base": 2,
"inner_exponent": 3,
"outer_exponent": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/power-of-a-power", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"base":2,"inner_exponent":3,"outer_exponent":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"base":2,"inner_exponent":3,"outer_exponent":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/power-of-a-power", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"base": 2,
"inner_exponent": 3,
"outer_exponent": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.power_of_a_power",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Limits
max_abs_base | 1000000000000000 |
max_abs_exponent | 1000000 |
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Non-finite inputs fail with invalid_input and are not billed.
What the power-of-a-power identity actually computes
The power-of-a-power rule is the exponent identity that turns a nested power into a single power of the same base: (a^m)^n equals a raised to the product of the two exponents, written a^(m·n). Zero raised to a non-positive simplified exponent is undefined in this contract, and a negative base with a non-integer simplified exponent is rejected when the IEEE real power is not finite.
Input contract, formula, and how errors are reported
If the product of the exponents is no longer finite, or if the evaluated power is Infinity or NaN, the task fails instead of returning a non-finite number. Failed validation is not billed. POST a JSON body with three required finite numbers: base, the common base a of the nested power; inner_exponent, the inner exponent m that sits on a inside the parentheses; and outer_exponent, the outer exponent n that raises the whole inner power. Each field may arrive as a JSON number or as a numeric string. Missing fields, empty strings, booleans, arrays, objects, NaN, and Infinity fail with invalid_input and the message names the offending field so a form can highlight it. On success the body echoes the three inputs, returns simplified_exponent as the finite product m times n, returns value as the finite real number obtained by raising the base to that product, and includes a short expression string plus the formula (a^m)^n = a^(m·n) so a page can print the identity next to the answer. Example: base two, inner_exponent three, and outer_exponent four yield simplified_exponent twelve and value four thousand ninety-six, with the expression (2^3)^4 = 2^12 = 4096. Pair the numeric fields with your own copy if you need a long classroom write-up of every rewrite; this endpoint computes the simplified exponent and the value and does not emit worked solutions beyond the formula and expression strings.
Technical questions
What does the power of a power calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.