Math

QuestionCalculate [1(216)2/3÷(127)4/3]1/2\left[\frac{1}{(216)^{-2 / 3}} \div\left(\frac{1}{27}\right)^{-4 / 3}\right]^{1 / 2}. Choose: (a) 2/32 / 3, (b) 1/31 / 3, (c) 1, (d) None.

Studdy Solution

STEP 1

Assumptions1. The expression is [1(216)/3÷(127)4/3]1/\left[\frac{1}{(216)^{- /3}} \div\left(\frac{1}{27}\right)^{-4 /3}\right]^{1 /}

STEP 2

First, we need to simplify the expression inside the brackets. Let's start with the first part of the expression 1(216)2/\frac{1}{(216)^{-2 /}}. We can simplify this by using the property an=1ana^{-n} = \frac{1}{a^n}.
1(216)2/=(216)2/\frac{1}{(216)^{-2 /}} = (216)^{2 /}

STEP 3

Next, we simplify the second part of the expression (127)/3\left(\frac{1}{27}\right)^{- /3}. We can simplify this by using the property an=1ana^{-n} = \frac{1}{a^n}.
(127)/3=(27)/3\left(\frac{1}{27}\right)^{- /3} = (27)^{ /3}

STEP 4

Now, we substitute these simplified expressions back into the original expression.
[(216)2/3÷(27)4/3]1/2\left[(216)^{2 /3} \div (27)^{4 /3}\right]^{1 /2}

STEP 5

We can simplify this further by using the property anbn=(ab)n\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n.
[(21627)2/3]1/2\left[\left(\frac{216}{27}\right)^{2 /3}\right]^{1 /2}

STEP 6

Now, we simplify the fraction inside the brackets.
[(81)2/3]1/2\left[\left(\frac{8}{1}\right)^{2 /3}\right]^{1 /2}

STEP 7

implify the expression inside the brackets.
[2/3]1/2\left[^{2 /3}\right]^{1 /2}

STEP 8

We can simplify this further by using the property (an)m=anm(a^n)^m = a^{n \cdot m}.
8(2/3)(1/2)8^{(2 /3) \cdot (1 /2)}

STEP 9

Calculate the exponent.
8/38^{ /3}

STEP 10

Calculate the value of 8/38^{ /3}.
8/3=28^{ /3} =2So, the solution to the expression [(216)2/3÷(27)4/3]/2\left[\frac{}{(216)^{-2 /3}} \div\left(\frac{}{27}\right)^{-4 /3}\right]^{ /2} is2, which corresponds to option (d) None of these.

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