Lesson 11 · from §94 of the 1910 text
Powers of Radicals
Powers of radicals
A power of a radical is a product of equal factors, so the rules of multiplication apply: \left(\sqrt[3]{5}\right)^2 = \sqrt[3]{25}, \qquad \left(3\sqrt[3]{2}\right)^3 = 27 \cdot 2 = 54, \qquad \left(\sqrt[4]{x}\right)^2 = \sqrt x.
Square or cube a binomial by the familiar formulas (a \pm b)^2 = a^2 \pm 2ab + b^2 and (a \pm b)^3 = a^3 \pm 3a^2b + 3ab^2 \pm b^3: \left(\sqrt5 - \sqrt2\right)^2 = 5 - 2\sqrt{10} + 2 = 7 - 2\sqrt{10}, \left(1 + \sqrt2\right)^3 = 1 + 3\sqrt2 + 3 \cdot 2 + 2\sqrt2 = 7 + 5\sqrt2.
Values. To find the value of an expression for a value of x that contains a radical, substitute and simplify. If x = 1 + \sqrt3, then x^2 - 2x - 2 = \left(4 + 2\sqrt3\right) - \left(2 + 2\sqrt3\right) - 2 = 0, so 1 + \sqrt3 is a root of x^2 - 2x - 2 = 0. Try 1 - \sqrt3 as well.
How to type radicals
- Square root:
sqrt(12)or√12; a coefficient goes in front:2sqrt(3)or2√3. - Cube root:
cbrt(5)or∛5. Any other root:root(x, 4)for \sqrt[4]{x},root(2, 6)for \sqrt[6]{2}. - Put the whole radicand in parentheses when it is more than one number or letter:
sqrt(2x),cbrt(a + b),sqrt(2 - sqrt(2)). - Fractional exponents use
^with parentheses:x^(2/3). - Letters under a radical sign stand for positive numbers, so \sqrt{x^2} = x.
The checker wants each answer in simplest form, with similar radicals collected and products of radicals under one sign. "Square" means find the square of the expression; "cube", its cube.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Square:
\sqrt[3]2
2\sqrt[3]3
2\sqrt[3]{12}
\sqrt{2 - \sqrt2}
3\sqrt{x + \sqrt3}
\left(2\sqrt[3]3\right)^2
2a\sqrt[3]{8x}
\sqrt[3]2 + \sqrt[3]3
\sqrt[3]{4 + 4\sqrt3}
\tfrac13\sqrt[3]{9 - 9\sqrt2}
\sqrt2 + \sqrt[3]3
2\sqrt[4]{3 - \sqrt2}
Cube:
2\sqrt[2]3
3\sqrt[2]2
\left(\sqrt2\right)^2
\left(\sqrt[2]3\right)^3
\sqrt3 - \sqrt6
2\sqrt2 + \sqrt3
\sqrt[3]3 - \sqrt[3]2
\sqrt{2 + \sqrt2}
\sqrt[3]2 - \sqrt3
Simplify:
R^2 + \left(\dfrac R2\sqrt{2 - \sqrt2}\right)^2
\sqrt{R^2 - \left(\dfrac R2\sqrt5 - \dfrac R2\right)^2}
\left[R^2 - \left(\dfrac{R\sqrt5 - R}{4}\right)^2\right]^{1/2}
\left[\left(R - \dfrac R2\sqrt3\right)^2 + \left(\dfrac R2\right)^2\right]^{1/2}
\left[R^2 - \left(\dfrac{R\sqrt{2 - \sqrt3}}{2}\right)^2\right]^{1/2}
\sqrt{R^2 - \left(\dfrac{R\sqrt{2 - \sqrt2}}{2}\right)^2}
\sqrt{\left(\dfrac{R\sqrt2}{2}\right)^2 + \left(R - \dfrac R2\sqrt2\right)^2}
\left(\dfrac R2\sqrt{2 + \sqrt2}\right)\left(\dfrac{R\sqrt{2 - \sqrt2}}{2}\right)4
Values
Find the value of x^2 + 4x + 1 if x = -2 + \sqrt3.
Find the value of x^2 - 2x - 3 if x = 2 + \sqrt5.
Find the values of x^2 - 4x - 1 if x = 2 \pm \sqrt5.
Find the values of 3x^2 + 3x - 5 if x = \tfrac12 \pm \tfrac12\sqrt{13}.
Do the values x = 2 \pm \sqrt3 satisfy the equation x^2 - 4x + 1 = 0?
Do the values x = -4 \pm \sqrt5 satisfy the equation x^2 + 8x + 11 = 0?
Exercises 80 and 81 suggest that when a + \sqrt b is a root of such an equation, so is a - \sqrt b. Given that 3 + \sqrt2 is a root of x^2 - 6x + 7 = 0, what is the other root? (Check it.)