Simplify using the exponent law. Express your answer using positive exponents only.
[4x^1/3]^1/2 [9x]^-3/2 / [27x]^-1/3
= 2(x^(1/6) ((1/27) x^(-3/2) (3) (x^(1/3)
= (6/27) x^(1/6-3/2+1/3)
= (2/9) x^-1
= 2/(9x)
To simplify the expression using the exponent law, we need to apply the rules for exponents. Let's break down the expression step by step:
First, let's simplify the expression inside the square brackets: [4x^(1/3)]^(1/2).
According to the exponent law, when we raise a power to another power, we multiply the exponents. Hence, we multiply (1/3) and (1/2) to simplify:
[4x^(1/3)]^(1/2) = 4^(1/2) * (x^(1/3))^(1/2)
Next, let's simplify 4^(1/2):
4^(1/2) is the square root of 4, which is 2. Therefore,
[4x^(1/3)]^(1/2) = 2 * (x^(1/3))^(1/2)
Now, let's simplify (x^(1/3))^(1/2):
According to the exponent law, when we multiply two exponents with the same base, we add the exponents. Therefore:
(x^(1/3))^(1/2) = x^((1/3) * (1/2)) = x^(1/6)
So, the expression inside the square brackets becomes:
[4x^(1/3)]^(1/2) = 2x^(1/6)
Now, let's simplify the expression inside the parentheses: [9x]^(-3/2).
According to the exponent law, when we have a negative exponent, we can move the term to the denominator and change the sign of the exponent. Therefore:
[9x]^(-3/2) = 1/[9x]^(3/2)
Lastly, let's simplify the expression in the denominator: [27x]^(-1/3).
Just like before, we move the term to the denominator and change the sign of the exponent:
[27x]^(-1/3) = 1/[27x]^(1/3)
Now, we have the following expression:
(2x^(1/6) * 1/[9x]^(3/2)) / (1/[27x]^(1/3))
To divide fractions, we can multiply the first fraction by the reciprocal of the second fraction. Therefore:
(2x^(1/6) * 1/[9x]^(3/2)) * ([27x]^(1/3)/1)
Now, let's apply the exponent law to simplify each term:
1/[9x]^(3/2) = 1/(9^(3/2) * (x^(3/2)) = 1/(9^(3/2) * x^(3/2))
[27x]^(1/3) = (27^(1/3) * (x^(1/3)) = 3x^(1/3)
Substituting these simplified expressions back into the main expression, we get:
(2x^(1/6) * 1/(9^(3/2) * x^(3/2))) * (3x^(1/3)/1)
To multiply with the same base, we add the exponents:
2 * x^(1/6 - 3/2 + 1/3)
Now, let's simplify the exponent:
1/6 - 3/2 + 1/3 = (1 - 9 + 2)/6 = -6/6 = -1
Therefore, the simplified expression is:
2x^(-1)
Expressing the answer using positive exponents, we move the term to the denominator and change the sign of the exponent:
2x^(-1) = 2 / x^1
Hence, the simplified expression with positive exponents is:
2/x