Algebra
Exponents and scientific notation
Properties of exponents
20 practice questions
2 video lessons
Theory + worked examples
Theory
The laws of exponents simplify products, quotients, and powers:
- Product: \(b^m b^n=b^{m+n}\).
- Quotient: \(\dfrac{b^m}{b^n}=b^{m-n}\).
- Power: \((b^m)^n=b^{mn}\).
- Product to a power: \((ab)^n=a^n b^n\).
Add to multiply, subtract to divide, multiply to raise a power.
The laws of exponents.
Worked examples.
The laws:
\[b^m b^n=b^{m+n},\quad \dfrac{b^m}{b^n}=b^{m-n},\quad (b^m)^n=b^{mn}\]
Same base required for the product and quotient rules.
How to apply the laws
- Multiplying same bases? Add exponents.
- Dividing? Subtract exponents.
- Power of a power? Multiply exponents.
- Raise each factor in a product.
Example 1 — Product rule
Simplify \(x^3\cdot x^2\).
Solution
Add the exponents.
| \(x^3\cdot x^2\) | \(=\) | \(x^{5}\) |
Example 2 — Quotient rule
Simplify \(\dfrac{x^5}{x^2}\).
Solution
Subtract the exponents.
| \(\dfrac{x^5}{x^2}\) | \(=\) | \(x^{3}\) |
Example 3 — Power rule
Simplify \((x^2)^3\).
Solution
Multiply the exponents.
| \((x^2)^3\) | \(=\) | \(x^{6}\) |
Example 4 — Product to a power
Simplify \((2x)^3\).
Solution
Raise each factor.
| \((2x)^3\) | \(=\) | \(2^3x^3=8x^3\) |
Common pitfalls
Add exponents to multiply, don't multiply them.
The base must be the same for product/quotient rules.
\((2x)^3=8x^3\) — the coefficient is cubed too.
Frequently asked questions
How do you multiply \(x^3\cdot x^2\)?
Add exponents: \(x^5\).
How do you simplify \((x^2)^3\)?
Multiply exponents: \(x^6\).
Do the bases need to match?
Yes for the product and quotient rules.
What is \((3x)^2\)?
\(9x^2\).
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