Short answer
Use exponent laws to combine like bases and rewrite roots as fractional exponents. Simplify radicals by factoring perfect powers from the radicand, and keep track of domain restrictions when even roots are involved.
- Products add exponents only when bases match.
- Negative exponents represent reciprocals, not negative values.
- Radicals simplify by extracting largest perfect-power factors.
Use exponent laws precisely
For the same base a, a^m * a^n = a^(m+n) and a^m / a^n = a^(m-n) when a is nonzero. A power raised to a power multiplies exponents: (a^m)^n = a^(mn).
Do not distribute exponents over addition. (a + b)^2 is not a^2 + b^2.
Connect radicals and rational exponents
The nth root of a can be written as a^(1/n). So sqrt(a) is a^(1/2), and cube root of a is a^(1/3). This notation helps when multiplying and dividing radical expressions.
- sqrt(50) = sqrt(25*2) = 5sqrt(2).
- a^(-3) = 1/a^3 for a not equal to zero.
- a^(m/n) means nth root of a^m.
Simplify an expression with exponents and radicals
Simplify (18x^5y^2)/(3x^2y) and then rewrite sqrt(x^6) for x >= 0.
- Divide coefficients: 18/3 = 6.
- Subtract exponents on like bases: x^(5-2) = x^3 and y^(2-1) = y.
- Simplified fraction is 6x^3y.
- For x >= 0, sqrt(x^6) = x^3 because (x^3)^2 = x^6.
Result: Results are 6x^3y and x^3 under the stated domain condition.
Common mistakes
- Adding exponents across unlike bases.
- Treating a negative exponent as making the value negative.
- Assuming sqrt(a+b) equals sqrt(a) + sqrt(b).
- Dropping domain conditions when simplifying even roots.
Try one
Simplify 27^(2/3).
27^(2/3) = (cube root of 27)^2 = 3^2 = 9.
Sources
- OpenStax: College AlgebraOpen textbook with full chapter coverage, worked examples, and practice problems.
- Khan Academy: Algebra and Algebra 2Official lessons and practice sets for core algebra skills.