mathematics

Seventh power

Seventh power is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Seventh power rather than just read about it. In short: In arithmetic and algebra, the seventh power of a number n is the result of multiplying seven instances of n together. So: n7 = n × n × n × n × n × n × n.

Key takeaways

Seventh power belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Seventh power to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Seventh power from memory before moving on to harder problems.

Reference excerpt

In arithmetic and algebra, the seventh power of a number n is the result of multiplying seven instances of n together. So:

n7 = n × n × n × n × n × n × n. Seventh powers are also formed by multiplying a number by its sixth power, the square of a number by its fifth power, or the cube of a number by its fourth power. The sequence of seventh powers of integers is:

0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 10000000, 19487171, 35831808, 62748517, 105413504, 170859375, 268435456, 410338673, 612220032, 893871739, 1280000000, 1801088541, 2494357888, 3404825447, 4586471424, 6103515625, 8031810176, ... (sequence A001015 in the OEIS) In the archaic notation of Robert Recorde, the seventh power of a number was called the "second sursolid".

Properties Leonard Eugene Dickson studied generalizations of Waring's problem for seventh powers, showing that every non-negative integer can be represented as a sum of at most 258 non-negative seventh powers (17 is 1, and 27 is 128). All but finitely many positive integers can be expressed more simply as the sum of at most 46 seventh powers. If powers of negative integers are allowed, only 12 powers are required. The smallest number that can be represented in two different ways as a sum of four positive seventh powers is 2056364173794800. The smallest seventh power that can be represented as a sum of eight distinct seventh powers is:

102 7 = 12 7 + 35 7 + 53 7 + 58 7 + 64 7 + 83 7 + 85 7 + 90 7 . {\displaystyle 102^{7}=12^{7}+35^{7}+53^{7}+58^{7}+64^{7}+83^{7}+85^{7}+90^{7}.}

The two known examples of a seventh power expressible as the sum of seven seventh powers are

568 7 = 127 7 + 258 7 + 266 7 + 413 7 + 430 7 + 439 7 + 525 7 {\displaystyle 568^{7}=127^{7}+258^{7}+266^{7}+413^{7}+430^{7}+439^{7}+525^{7}} (M. Dodrill, 1999); and

626 7 = 625 7 + 309 7 + 258 7 + 255 7 + 158 7 + 148 7 + 91 7 {\displaystyle 626^{7}=625^{7}+309^{7}+258^{7}+255^{7}+158^{7}+148^{7}+91^{7}} (Maurice Blondot, 11/14/2000); any example with fewer terms in the sum would be a counterexample to Euler's sum of powers conjecture, which is currently only known to be false for the powers 4 and 5.

See also Eighth power Sixth power Fifth power (algebra) Fourth power Cube (algebra) Square (algebra)

References

Worked examples

Example 1 — a first encounter with Seventh power

Start with the simplest possible case. Write down what Seventh power claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Seventh power before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Seventh power ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Seventh power

In research: Seventh power appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Seventh power in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Seventh power is common in secondary-school and first-year university syllabi. It links to neighbouring topics Elementary arithmetic, Figurate numbers, Integer sequences, so understanding it makes those chapters shorter.
In everyday life: Look for Seventh power outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Seventh power in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Seventh power means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Seventh power out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Seventh power in simple terms?

In arithmetic and algebra, the seventh power of a number n is the result of multiplying seven instances of n together. So: n7 = n × n × n × n × n × n × n.

Why does Seventh power matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Seventh power?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Seventh power.