SOLUTION: • 5 times the 4th root of x to the 6th power plus 3 times the 4th root of 81 times x to the 8th power. How do you write this and solve it.

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Question 392265: • 5 times the 4th root of x to the 6th power plus 3 times the 4th root of 81 times x to the 8th power. How do you write this and solve it.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Did you post the entire question exactly as it was given to you? I ask because there are two ways to interpret what you posted. It could be written as:
5%2Aroot%284%2C+x%5E6%29%2B3%2Aroot%284%2C+81x%5E8%29
or
%285%2Aroot%284%2C+x%29%29%5E6%2B3%2Aroot%284%2C+81x%5E8%29
depending on whether "to the 6th power" applies to just the "x" (the first expression) or to 5%2Aroot%284%2C+x%29 (the second expression). Without any further information I would guess that the first expression is correct and that is the one I will simplify.

Important: Although your post does not mention it, many times these problems are given with a statement to the effect that the variables are to assumed to have positive (or non-negative) values. If there was such a statement in your problem, then all uses of absolute value below are not necessary and can be removed!

As the expression is written, the two terms are not like terms. So we cannot add them together. But the radicands (the expression within a radical is called a radicand) of each of the two 4th roots have factors that are powers of 4. So the two radicals can be simplified.

Simplifying these radicals starts with factoring each radicand into factors where at least one factor is a power of 4:
5%2Aroot%284%2C+x%5E4%2Ax%5E2%29%2B3%2Aroot%284%2C+81%2A%28x%5E2%29%5E4%29
Since 81+=+3%5E4 we can write this as:
5%2Aroot%284%2C+x%5E4%2Ax%5E2%29%2B3%2Aroot%284%2C+3%5E4%2A%28x%5E2%29%5E4%29
Next we use the property of radicals, root%28a%2C+p%2Aq%29+=+root%28a%2C+p%29%2Aroot%28a%2C+q%29, to separate the power of 4 factors into their own 4th roots. (Any factors that are not powers of 4 are put into their own 4th root, too.):

Each of the 4th roots of powers of 4 will simplify.
5%2Aabs%28x%29%2Aroot%284%2C+x%5E2%29%2B3%2A3%2Ax%5E2%29
which simplifies to:
5%2Aabs%28x%29%2Aroot%284%2C+x%5E2%29%2B9x%5E2%29
Note the use of absolute value. There are two 4th roots of any positive number, a positive 4th root and a negative 4th root. The positive 4th root of q is written as root%284%2C+q%29 and the negative 4th root of q is written as -root%284%2C+q%29. Your original expression referred to the positive 4th roots in both terms! So both terms were positive (or at least non-negative). After we simplify the terms should still be positive.

But we don't know if "x" is positive or negative. This is why we use absolute value in the first term. Using absolute value ensures that the first term remains positive (or at least non-negative).

Absolute value is not needed in the second term because the x%5E2 will never be negative. So we don't need to do anything to make sure it won't be negative.

Note: The expression above may be the desired answer. However there is a further simplification that can be done that you may have not yet learned. If you haven't learned fractional exponents and how they can be used to simplify radical expressions, then we are finished.

The remaining radical can still be simplified if we use fractional exponents to express the radical. In general root%28p%2C+z%5Eq%29 can be written as z%5E%28%28q%2Fp%29%29. Using this pattern on the remaining radical we get:
5%2Aabs%28x%29%2Ax%5E%28%282%2F4%29%29%2B9x%5E2%29
Looking at the exponent we should see that it can be reduced:
5%2Aabs%28x%29%2Ax%5E%28%281%2F2%29%29%2B9x%5E2%29
We can rewrite this back in radical form:
5%2Aabs%28x%29%2Aroot%282%2C+abs%28x%5E1%29%29%2B9x%5E2%29
or
5%2Aabs%28x%29%2Asqrt%28abs%28x%29%29%2B9x%5E2%29
Note the use of absolute value again. The radicands of even-numbered roots like square roots and 4th roots may not be negative. Up to now the radicands have had even powers of x (which cannot be negative no matter what value "x" has). But now we have a radicand that is just "x". We still don't know if "x" is positive or negative (or zero). So we use the absolute value on "x" to make sure that the radicand of the square root is not negative.