Question 4 of 28intermediate🔧 ApplyShort Answer3 marks

Write the given number as the product of two or more powers in three different ways. The powers can be any integers. (i) 64364^3 (ii) 1928192^8 (iii) 32−532^{-5}

(i)
Answer

Three ways: 643=86=49=21864^3 = 8^6 = 4^9 = 2^{18}

(ii)
Answer

Three ways: 1928=(1924)2=(1922)4=(1921)8192^8 = (192^4)^2 = (192^2)^4 = (192^1)^8

(iii)
Answer

Three ways: 32−5=(32−1)5=(32−5)1=2−2532^{-5} = (32^{-1})^5 = (32^{-5})^1 = 2^{-25}

Explanation

(i) Since 64=26=82=4364 = 2^6 = 8^2 = 4^3, we have 643=(26)3=21864^3 = (2^6)^3 = 2^{18}, 643=(82)3=8664^3 = (8^2)^3 = 8^6, 643=(43)3=4964^3 = (4^3)^3 = 4^9 (ii) Using (na)b=nab(n^a)^b = n^{ab}, we can write 1928=(1924)2192^8 = (192^4)^2 since 4×2=84 \times 2 = 8, or (1922)4(192^2)^4 since 2×4=82 \times 4 = 8 (iii) Since 32=2532 = 2^5, we have 32−5=(25)−5=2−2532^{-5} = (2^5)^{-5} = 2^{-25}. Also 32−5=(32−1)532^{-5} = (32^{-1})^5

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