Question 405407
Please use parentheses to make things clear. I cannot tell if the exponent of 8 applies to the (x+3) or to the sqrt(x+3). So it could be
{{{sqrt((x+3)^8)}}}
or
{{{(sqrt(x+3))^8}}}
These expressions are different. Tutors are more likely to help when the problem is clear. (This might be why you have gotten no responses in two days.)<br>
In this particular case, both expression simplify to the same thing so I will go ahead and solve the problem. One way to simplify this is to use a fractional exponent for the square root:
{{{((x+3)^8)^(1/2)}}}
or
{{{((x+3)^(1/2))^8}}}
When raising a power to a power the rule is to multiply the exponents. And since both 8 * 1/2 and 1/2 * 8 are both 4, both of the expressions above work out to be:
{{{(x+3)^4}}}
And since something to the 4th power cannot be negative, absolute value is not needed.<br>
Simplifying with the radical form is a little more difficult. I'll do each of the possible expressions separately.
{{{sqrt((x+3)^8)}}}
To simplify this square root we factor the radicand (the expression inside) into perfect squares:
{{{sqrt((x+3)^2*(x+3)^2*(x+3)^2*(x+3)^2)}}}
Then we use a property of radicals, {{{root(a, p*q) = root(a, p)*root(a, q)}}}, to split the square root of a product into a product of the square roots fo the factors:
{{{sqrt((x+3)^2)*sqrt((x+3)^2)*sqrt((x+3)^2)*sqrt((x+3)^2))}}}
Each of the square roots simplify. In this case, since we do not know if x+3 is positive, negative or zero, we use absolute values:
{{{abs(x+3)*abs(x+3)*abs(x+3)*abs(x+3)}}}
or
{{{(abs(x+3))^4}}}
Again, since anything to the 4th power will never be negative, we can remove the absolute value:
{{{(x+3)^4}}}<br>
For
{{{(sqrt(x+3))^8}}}
Let's start by writing it out without the exponent of 8:
{{{sqrt(x+3)*sqrt(x+3)*sqrt(x+3)*sqrt(x+3)*sqrt(x+3)*sqrt(x+3)*sqrt(x+3)*sqrt(x+3)}}}
Since multiplication is Associative we can group these as we choose:
{{{(sqrt(x+3)*sqrt(x+3)) * (sqrt(x+3)*sqrt(x+3)) * (sqrt(x+3)*sqrt(x+3)) * (sqrt(x+3)*sqrt(x+3))}}}
Each grouped pair of factors simplifies. (Note: since the original radicand of this expression was x+3 we know that x+3 cannot be negative. Therefore, aboslute values will not be needed on this expression. In the other possible expression, {{{sqrt((x+3)^8)}}}, the radicand is {{{(x+3)^8}}}. With {{{(x+3)^8}}}, the x+3 can be anything, including negative, because after raise it to the 8th power it never be negative. Since the x+3 in {{{sqrt((x+3)^8)}}} could be negative, we needed the absolute value, at least temporarily.) So now we have:
{{{(x+3)(x+3)(x+3)(x+3)}}}
or
{{{(x+3)^4}}}<br>
So both {{{sqrt((x+3)^8)}}} and {{{(sqrt(x+3))^8}}} simplify to {{{(x+3)^4}}} no matter which way you simplify it.