Laws of Indices Simplifier
Simplify same-base multiplication/division, power of a power, negative and fractional exponents.
Input Data
Results
At a glance:Index laws: same-base multiply b^m·b^n=b^(m+n); divide b^m/b^n=b^(m−n); power of a power (b^m)^n=b^(mn). Negative exponent b^(−m)=1/b^m; fractional exponent b^(m/n)=ⁿ√(b^m). The tool merges exponents and shows the law used.
Formula
b^m · b^n = b^(m+n).
b^m / b^n = b^(m−n).
(b^m)^n = b^(m·n).
b^(−m)=1/b^m; b^(m/n)=ⁿ√(b^m).
$$b^{m}\cdot b^{n}=b^{m+n}$$$$\frac{b^{m}}{b^{n}}=b^{m-n}$$$$(b^{m})^{n}=b^{m n}$$$$b^{-m}=\frac{1}{b^{m}},\quad b^{m/n}=\sqrt[n]{b^{m}}$$How to Use
- Enter base b (variable or number).
- Pick an operation.
- Enter exponents m (and n); the tool merges them and states the law.
- If the base is numeric, the numeric value is also shown.
Index laws reference
| Operation | Rule | Example |
|---|---|---|
| Multiply | b^m·b^n=b^(m+n) | x²·x³=x⁵ |
| Divide | b^m/b^n=b^(m−n) | x⁵/x²=x³ |
| Power | (b^m)^n=b^(mn) | (x²)³=x⁶ |
| Negative | b^(−m)=1/b^m | 2^(−1)=1/2 |
| Fractional | b^(m/n)=ⁿ√(b^m) | 4^(1/2)=2 |
Division and negative exponents require b≠0.
Case Studies
x³ · x²
Add exponents: 3+2=5 → x^5.
2^(−3)
Negative exponent: 1/2³ = 1/8 = 0.125.
8^(2/3)
Cube root of 8 is 2, squared is 4.
FAQ
What is a negative exponent?
b^(−m) means 1/b^m — take the reciprocal and make the exponent positive. E.g. 2^(−2)=1/4.
How to evaluate a fractional exponent?
b^(m/n) is the n-th root of b^m (or b to the 1/n, then to the m). E.g. 8^(2/3)=(³√8)²=2²=4.
Can different bases be merged?
Index laws apply only to the same base. Different bases (x²·y³) cannot merge exponents.
Why does (b^m)^n multiply exponents?
It is n copies of b^m, i.e. n×m copies of b, so b^(mn).
Related Tools
References
Content review: Calculatorism Science Team. The three index laws plus negative and fractional exponent conversion have been verified. Results are for reference only.