Understanding nth Roots | principal nth root
Supposeweknowthat[latex]{a}{3}=8[/latex].Wewanttofindwhatnumberraisedtothe3rdpowerisequalto8.Since[latex]{2}{3}=8[/latex],wesaythat2isthecuberootof8.Thenthrootof[latex]a[/latex]isanumberthat,whenraisedtothenthpower,gives[latex]a[/latex].Forexample,[latex]-3[/latex]isthe5throotof[latex]-243[/latex]because[latex]{left(-3ight)}{5}=-243[/latex].If[latex]a[/latex]isarealnumberwithatleastonenthroot,thentheprincipalnthrootof[latex]a[/latex]isthenumberwiththesamesignas[latex]a[/latex]that,whenraised...
Suppose we know that [latex]{a}{3}=8[/latex]. We want to find what number raised to the 3rd power is equal to 8. Since [latex]{2}{3}=8[/latex], we say that 2 is the cube root of 8.
The nth root of [latex]a[/latex] is a number that, when raised to the nth power, gives [latex]a[/latex]. For example, [latex]-3[/latex] is the 5th root of [latex]-243[/latex] because [latex]{left(-3 ight)}{5}=-243[/latex]. If [latex]a[/latex] is a real number with at least one nth root, then the principal nth root of [latex]a[/latex] is the number with the same sign as [latex]a[/latex] that, when raised to the nth power, equals [latex]a[/latex].
The principal nth root of [latex]a[/latex] is written as [latex]sqrt[n]{a}[/latex], where [latex]n[/latex] is a positive integer greater than or equal to 2. In the radical expression, [latex]n[/latex] is called the index of the radical.
A General Note: Principal nth RootIf [latex]a[/latex] is a real number with at...