11
Simplifying algebraic expressions with rational exponents
Textbook: pp. 49–55
GoalSimplify algebraic expressions with rational exponents using short multiplication formulas and power properties.
New words
simplifying an expression · ifodani soddalashtirishshort multiplication formula · qisqa ko‘paytirish formulasitaking out a common factor · umumiy ko‘paytuvchini chiqarishreducing · qisqartirish
Explanation
In such expressions the letters are taken to be positive. Short multiplication formulas work for these powers too: put x^(1/2) = a and y^(1/2) = b and use (a ± b)² = a² ± 2ab + b² and a² - b² = (a - b)(a + b). For example, x - y = (x^(1/2) - y^(1/2))(x^(1/2) + y^(1/2)). Before reducing a fraction, take the common factor out of the numerator and denominator. When multiplying, exponents add; when dividing they subtract; when raising to a power they multiply. Write the answer in the simplest form, as a root or a power.
Worked examples
(a - b)/(a^(1/2) - b^(1/2)) = (a^(1/2) - b^(1/2))(a^(1/2) + b^(1/2))/(a^(1/2) - b^(1/2)) = a^(1/2) + b^(1/2). For a = 16, b = 9 the value is 4 + 3 = 7.
(x^(2/3) - 9)/(x^(1/3) + 3) = (x^(1/3) - 3)(x^(1/3) + 3)/(x^(1/3) + 3) = x^(1/3) - 3; (x^(1/2) + y^(1/2))² = x + 2(xy)^(1/2) + y.
Class activity
“Substitution trick”: pupils replace x^(1/2) by the letter a in a complicated expression, simplify, and then return to x.
Practice
1
Simplify a^(1/2) · a^(3/2).
a²
2
Simplify (a^(1/2) + 2)(a^(1/2) - 2).
a - 4
3
Simplify (x - 1)/(x^(1/2) + 1) and find its value at x = 25.
x^(1/2) - 1; 4
4
Why are the letters taken to be positive in such expressions?
Because powers like a^(1/2) make sense for a ≥ 0, and the power properties are proved for positive bases.