document.write( "Question 396287: 3(x+1)^(4/3)=48 \n" ); document.write( "

Algebra.Com's Answer #281148 by jsmallt9(3758) $\"\"$ $\"About$

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$\"3%28x%2B1%29%5E%284%2F3%29=48\"$
\n" ); document.write( "First let's isolate the base, (x+1), and its exponent, 4/3, on one side of the equation by dividing both sides by 3:
\n" ); document.write( " $\"%28x%2B1%29%5E%284%2F3%29=16\"$
\n" ); document.write( "Next we want to find a way to change the exponent from 4/3 to a 1. We can do so by combining all of the following ideas:

One is allowed to raise both sides of an equation to any non-zero power.
The rule for exponents when raising a power to a power is to multiply the exponents.
Multiplying reciprocals always results in a 1.

\n" ); document.write( "All of this tells us that if we raise each side of the equation to the reciprocal of 4/3 power, we will end up with an exponent of 1 on the left side. The reciprocal of 4/3 is 3/4:
\n" ); document.write( " $\"%28%28x%2B1%29%5E%284%2F3%29%29%5E%283%2F4%29=%2816%29%5E%283%2F4%29\"$
\n" ); document.write( "On the left side the exponent turns into a 1, as planned:
\n" ); document.write( " $\"%28x%2B1%29%5E1=%2816%29%5E%283%2F4%29\"$
\n" ); document.write( "Simplifying the left side is simple. The right side is not so simple. If you have trouble with negative or fractional exponents I find it can be helpful to factor the exponent in a certain way:

If the exponent is negative, factor out a -1.
If the exponent is fractional and if the numerator is not a 1, then factor out the numerator. (You'll see what this means in a moment.)

\n" ); document.write( "The exponent on the right is not negative but it is a fraction whose numerator is not a 1. So we will factor out the 3:
\n" ); document.write( " $\"x%2B1+=+%2816%29%5E%283%2A%281%2F4%29%29\"$
\n" ); document.write( "Now, looking at the exponent in factored form, the 3 tells us we will be cubing and the 1/4 tells us that we will be finding a 4th root. And we can do these two operations in any order! Normally 4th roots aren't simple. However, since $\"16+=+2%5E4\"$ , finding the 4th root of 16 is rather easy. So we will start with the 4th root and then cube it:
\n" ); document.write( " $\"x%2B1+=+%2816%29%5E%28%281%2F4%29%2A3%29\"$
\n" ); document.write( " $\"x%2B1+=+%2816%5E%281%2F4%29%29%5E3\"$
\n" ); document.write( " $\"x%2B1+=+%282%29%5E3\"$
\n" ); document.write( "x+1 = 8
\n" ); document.write( "Now we just subtract 1:
\n" ); document.write( "x = 7 \n" ); document.write( "

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