Algebra
Exponents and scientific notation
Exponents
20 practice questions
2 video lessons
Theory + worked examples
Theory
An exponent counts repeated multiplication of a base:
\[b^n=\underbrace{b\cdot b\cdots b}_{n\text{ factors}}.\]
- \(b^0=1\) for any nonzero \(b\).
- \(b^{-n}=\dfrac{1}{b^n}\) — a reciprocal.
The exponent applies to the base only unless the base is in parentheses.
Exponents as repeated multiplication.
Worked examples.
Key facts:
\[b^0=1,\qquad b^{-n}=\dfrac{1}{b^n}\]
\(-2^2=-4\) but \((-2)^2=4\) — parentheses matter.
How to evaluate a power
- Identify the base and exponent.
- Multiply the base that many times.
- A zero exponent gives \(1\).
- A negative exponent gives a reciprocal.
Example 1 — Expand a power
Evaluate \(2^4\).
Solution
Multiply four \(2\)'s.
| \(2^4\) | \(=\) | \(2\cdot2\cdot2\cdot2\) |
| \(=\) | \(16\) |
Example 2 — Negative base
Evaluate \((-2)^3\).
Solution
An odd power keeps the negative.
| \((-2)^3\) | \(=\) | \(-8\) |
Example 3 — Zero exponent
Evaluate \(5^0\).
Solution
Any nonzero base to the zero is \(1\).
| \(5^0\) | \(=\) | \(1\) |
Example 4 — Negative exponent
Evaluate \(2^{-2}\).
Solution
A negative exponent means reciprocal.
| \(2^{-2}\) | \(=\) | \(\dfrac{1}{2^2}=\dfrac14\) |
Common pitfalls
\((-2)^2=4\), but \(-2^2=-4\) — watch parentheses.
\(b^0=1\), not \(0\).
A negative exponent is a reciprocal, not a negative number.
Frequently asked questions
What is an exponent?
The number of times the base is multiplied by itself.
What is \(b^0\)?
\(1\), for any nonzero base.
What does a negative exponent mean?
A reciprocal: \(b^{-n}=\dfrac1{b^n}\).
Is \(-3^2=9\)?
No — \(-3^2=-9\); only \((-3)^2=9\).
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