Math

QuestionMatch the expression: 27((33)1)27 \cdot\left(\left(3^{3}\right)^{-1}\right)

Studdy Solution

STEP 1

Assumptions1. The expression is 27((33)1)27 \cdot\left(\left(3^{3}\right)^{-1}\right). The exponent rules apply, where (am)n=amn(a^{m})^{n} = a^{m \cdot n} and an=1/ana^{-n} =1/a^{n}

STEP 2

First, we simplify the expression inside the parentheses. According to the exponent rules, (am)n=amn(a^{m})^{n} = a^{m \cdot n}.
()1=1\left(^{}\right)^{-1} =^{ \cdot -1}

STEP 3

Now, simplify the exponent by multiplying 33 by 1-1.
331=333^{3 \cdot -1} =3^{-3}

STEP 4

Next, apply the exponent rule an=1/ana^{-n} =1/a^{n} to simplify 333^{-3}.
33=1/333^{-3} =1/3^{3}

STEP 5

Calculate the value of 333^{3}.
33=273^{3} =27

STEP 6

Substitute 333^{3} with 2727 in the expression 1/331/3^{3}.
1/33=1/271/3^{3} =1/27

STEP 7

Now, substitute (33)1\left(3^{3}\right)^{-1} with 1/271/27 in the original expression.
27((33)1)=271/2727 \cdot\left(\left(3^{3}\right)^{-1}\right) =27 \cdot1/27

STEP 8

Calculate the value of 271/2727 \cdot1/27.
271/27=127 \cdot1/27 =1The value of the expression 27((33)1)27 \cdot\left(\left(3^{3}\right)^{-1}\right) is 11.

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