Question 395159
{{{3*root(4, 24)*5*root(4, 2)}}}
This is all multiplication. So we can use the Commutative and Associative Properties to rearrange the order and grouping in any way we choose:
{{{(3*5)*(root(4, 24)*root(4, 2))}}}
To multiply the radicals we use the property of all radicals: {{{root(a, p)*root(a, q) = root(a, p*q)}}}:
{{{15*root(4, 24*2)}}}
or
{{{15*root(4, 48)}}}
Just like fractions should be reduced, radicals should be simplified. Simplifying radicals involves finding factors of the radicand (the expression within a radical) that are powers of the type of root, if any. In this problem with its 4th root, we are looking for factors of 48 that are a power of 4. Since {{{2^4 = 16}}} and since 16 is a factor of 48, this radical will simplify. We start by writing the radicand in factored form:
{{{15*root(4, 16*3)}}}
Then we use the property of radicals we used earlier. Only this time we use it in the other direction: to split a single radical into a product of radicals. By doing this we get the power of 4 factor into its own radical:
{{{15*root(4, 16)*root(4, 3)}}}
The 4th root of 16 is 2 so this becomes:
{{{15*2*root(4, 3)}}}
which simplifies to
{{{30*root(4, 3)}}}