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Mathematics

Express each of the following in exponential notation :

(i) 343512\dfrac{343}{512}

(ii) −32243\dfrac{-32}{243}

(iii) −1128\dfrac{-1}{128}

(iv) 72964\dfrac{729}{64}

Exponents

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Answer

(i) 343512\dfrac{343}{512}

We have:

343 = 7 x 7 x 7 = 73

512 = 8 x 8 x 8 = 83

∴ 343512=7383=(78)3\dfrac{343}{512} = \dfrac{7^3}{8^3} = \Big(\dfrac{7}{8}\Big)^3

Hence, the answer is (78)3\Big(\dfrac{7}{8}\Big)^3

(ii) −32243\dfrac{-32}{243}

We have:

-32 = (-2) x (-2) x (-2) x (-2) x (-2) = (-2)5

243 = 3 x 3 x 3 x 3 x 3 = 35

∴ −32243=(−2)535=(−23)5\dfrac{-32}{243} = \dfrac{(-2)^5}{3^5} = \Big(\dfrac{-2}{3}\Big)^5

Hence, the answer is (−23)5\Big(\dfrac{-2}{3}\Big)^5

(iii) −1128\dfrac{-1}{128}

We have denominator 128:

∴ 128 = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 27

-1 = (-1)7 \quad (Since any odd power of -1 is -1)

∴ −1128=(−1)727=(−12)7\dfrac{-1}{128} = \dfrac{(-1)^7}{2^7} = \Big(\dfrac{-1}{2}\Big)^7

Hence, the answer is (−12)7\Big(\dfrac{-1}{2}\Big)^7

(iv) 72964\dfrac{729}{64}

We have:

729 = 3 x 3 x 3 x 3 x 3 x 3 = 36

64 = 2 x 2 x 2 x 2 x 2 x 2 = 26

∴ 72964=(32)6\dfrac{729}{64} = \Big(\dfrac{3}{2}\Big)^6

Hence, the answer is (32)6\Big(\dfrac{3}{2}\Big)^6

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