An Exponent Theorem

Let us consider the statement $b^n - b^{n-1} = b^{n-1} (b-1)$.

By quotient property of exponents,

$b^n \div b^{n-1} = b$

Subtracting 1 from both sides,

$(b^n \div b^{n-1}) - 1 = b - 1$

$(b^n \div b^{n-1}) - (b^{n-1} \div b^{n-1}) = b - 1$

$(b^n - b^{n-1}) \div b^{n-1} = b - 1$

Multiplying both sides by $b^{n-1}$ ,

$\boxed{b^n - b^{n-1} = b^{n-1} (b-1)}$

Let $b = \big\{x | x \in \mathbb{R}\big\}$ .

If $b = 0$ , let $n - 1 = \big\{x | x \in (1,+\infty)\big\}$ .

#NumberTheory #Exponents #Theorem

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Note that you have not proven the theorem. You have only demonstrated that it is true in 5 specific cases.

Calvin Lin Staff - 5 years, 9 months ago

Thank you for commenting your thoughts on this. This is my first note on Brilliant. Thank you again.

Adriel Padernal - 5 years, 9 months ago

You can use the property of the exponents. Provided that $b \neq 0$ , $b^n-b^{n-1} = b^{n-1}(\frac{b^n}{b^{n-1}}-\frac{b^{n-1}}{b^{n-1}})$ $b^n-b^{n-1} = b^{n-1}(b^{n-(n-1)}-b^{(n-1)-(n-1)})$ $b^n-b^{n-1} = b^{n-1}(b^1-b^0) = b^{n-1}(b-1)$ $Q.E.D.$

Kay Xspre - 5 years, 9 months ago

Thank you very much, I have now provided more supportive evidence for my theorem.

Adriel Padernal - 5 years, 7 months ago

Kay's comment isn't just "supportive evidence". It is a proof of the statement which you stated.

Calvin Lin Staff - 5 years, 7 months ago

@Calvin Lin – Thank you, I understand. At first, I was doubtful in a way that others might correct me for calling it "proof" so I decided to call it "supportive evidence."

Adriel Padernal - 5 years, 7 months ago

@Adriel Padernal – Thanks for updating the note!

It can take a while for you to become familiar with mathematical rigor. We all have to start somewhere, and I'm glad that you're learning!

Calvin Lin Staff - 5 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$