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QuestionНамери стойността на израза 913+(−3)25+(−27)8816−4⋅(−3)23\frac{9^{13}+(-3)^{25}+(-27)^{8}}{81^{6}-4 \cdot(-3)^{23}}.

Studdy Solution

STEP 1

Assumptions1. The expression is 913+(−3)25+(−27)8816−4⋅(−3)23\frac{9^{13}+(-3)^{25}+(-27)^{8}}{81^{6}-4 \cdot(-3)^{23}} . We need to simplify this expression

STEP 2

We can simplify the expression by breaking down the bases of the powers into their prime factors.
9 =^2,81 =^4, -27 = -^

STEP 3

Substitute these values into the expression.
\frac{(3^2)^{13}+(-3)^{25}+(-3^3)^{8}}{(3^)^{6}- \cdot(-3)^{23}}

STEP 4

implify the powers of powers by multiplying the exponents.
326+(−3)25+(−3)24324−4⋅(−3)23\frac{3^{26}+(-3)^{25}+(-3)^{24}}{3^{24}-4 \cdot(-3)^{23}}

STEP 5

Notice that (−3)25(-3)^{25} and (−3)23(-3)^{23} are odd powers of -3, which will be negative. So we can rewrite the expression as326−325−324324+4⋅323\frac{3^{26}-3^{25}-3^{24}}{3^{24}+4 \cdot3^{23}}

STEP 6

Factor out 3243^{24} from each term in the numerator and denominator.
324(32−3−1)324(1+4⋅3−1)\frac{3^{24}(3^2-3-1)}{3^{24}(1+4 \cdot3^{-1})}

STEP 7

implify the expressions in the parentheses.
324(9−3−1)324(1+4/3)\frac{3^{24}(9-3-1)}{3^{24}(1+4/3)}

STEP 8

Calculate the values inside the parentheses.
324(5)324(4/3+1)\frac{3^{24}(5)}{3^{24}(4/3+1)}

STEP 9

implify the expression in the denominator.
324(5)324(7/3)\frac{3^{24}(5)}{3^{24}(7/3)}

STEP 10

implify the fraction by dividing the numerator and denominator by 3243^{24}.
57/3\frac{5}{7/3}

STEP 11

Divide5 by7/3 to get the final answer.
57/3=5⋅37=157\frac{5}{7/3} = \frac{5 \cdot3}{7} = \frac{15}{7}The value of the expression is 157\frac{15}{7}.

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