Section 1.6 - More Equations: Odd roots, Radicals, Rational Exponents, Quadratic in form

Section Objectives

1. Solve equations using odd roots.
2. Solve radical equations.
3. Solve equations involving rational exponents.
4. Solve equations that are quadratic in form.

Examples Involving Odd Roots

• Solve for $w$: $\quad w^3=-8$

• Solve for $r$: $\quad r^3 = 25$

• Solve for $y$: $\quad (y+8)^3-16=0$

Radical Equations

Radical equations are equations involving radicals and roots. A typical approach for solving certain kinds of radical equations is to isolate the radical and "undo" it by raising each side of the equation to the appropriate power. It is crucial that you check your answers in the original problem.

Keep in mind that radicals with even-power indicies can never be negative!

Examples with Radicals

• Solve for $w$: $\quad \sqrt{w}=-6$

• Solve for $u$: $\quad \sqrt{u} - 14=0$

• Solve for $x$: $\quad -1+\sqrt{x-12}=7$

• Solve for $r$: $\quad 2=\sqrt{3r-12}-1$

• Solve for $u$: $\quad u = \sqrt{5u+14}$

• Solve for $w$: $\quad \sqrt[4]{w}=-4$

• Solve for $t$: $\quad \sqrt[3]{t}=2$

Equations with Rational Exponents

A rational exponent is an exponent that can be written as a fraction involving whole numbers. Rational (fractional) exponents denotes radicals, and expressions involving fractional exponents can be written in radical form. The key idea is that

$\displaystyle x^{n/m} = \sqrt[m]{x^n} = \left( \sqrt[m]{x} \right)^n$.

Be careful about making sure that these kinds of expressions are defined!

For example, $x^{1/2} = \sqrt{x}$ is only defined when $x \ge 0$. All other $x$-values are restricted.

On the other hand, $x^{1/3} = \sqrt[3]{x}$ is defined for all real $x$.

To solve an equation involving rational exponents, we might

1. rewrite the equation with radicals and solve as above, or
2. use "undoing" by raising each side to the appropriate $m/n$ power.

Examples with Rational Exponents

• Solve for $w$: $\quad w^{1/4} = -5$

• Solve for $r$: $\quad r^{1/4}=3$

• Solve for $u$: $\quad (6u+4)^{1/3} + 3 = 7$

• Solve for $x$: $\quad (3x+1)^{1/4}+5=3$

• Solve for $z$: $\quad (z+2)^{3/2} = 3$

• Solve for $x$: $\quad (x+7)^{2/5} = -1$

Examples that are Quadratic in Form

Some equations can be reduced to simpler equations by means of a substitution of variables. Each of these can be reduced to quadratic and an appropriate substitution.

• Solve for $y$: $\quad (y^2-11)^2-10(y^2-11)+25=0$

• Solve for $x$: $\quad x^4-37x^2+36=0$

• Solve for $w$: $\quad 2w^{2/3}=3w^{1/3}+20$