Express each of the following in exponential notation :(i) (5/21)^3 x (5/21)^8 (ii) (-7/3)^11 x (-7/3)^13 (iii) (13/43)^7 / (13/43)^2 (iv) (-16/35)^16 / (-16/35)^3 (v) (-7/15)^12 / (-7/15)^15 (vi) (1/24)^13 / (1/24)^16

(i) Using the multiplication rule, we add the exponents: 3 + 8 = 11. Hence, the answer is (ii) Using the multiplication rule, we add the exponents: 11 + 13 = 24. Hence, the answer is (iii) Using the division rule, we subtract the exponents: 7 - 2 = 5. Hence, the answer is (iv) Using the division rule, we subtract the exponents: 16 - 3 = 13. Hence, the answer is (v) Using the division rule, we subtract the exponents: 12 - 15 = -3. Hence, the answer is (vi) Using the division rule, we subtract the exponents: 13 - 16 = -3. Hence, the answer is 243

Mathematics

Express each of the following in exponential notation :
(i) $\Big(\dfrac{5}{21}\Big)^3 \times \Big(\dfrac{5}{21}\Big)^8$
(ii) $\Big(\dfrac{-7}{3}\Big)^{11} \times \Big(\dfrac{-7}{3}\Big)^{13}$
(iii) $\Big(\dfrac{13}{43}\Big)^7 ÷ \Big(\dfrac{13}{43}\Big)^2$
(iv) $\Big(\dfrac{-16}{35}\Big)^{16} ÷ \Big(\dfrac{-16}{35}\Big)^3$
(v) $\Big(\dfrac{-7}{15}\Big)^{12} ÷ \Big(\dfrac{-7}{15}\Big)^{15}$
(vi) $\Big(\dfrac{1}{24}\Big)^{13} ÷ \Big(\dfrac{1}{24}\Big)^{16}$

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(i) $\Big(\dfrac{5}{21}\Big)^3 \times \Big(\dfrac{5}{21}\Big)^8$

Using the multiplication rule, we add the exponents: 3 + 8 = 11.

$\therefore \Big(\dfrac{5}{21}\Big)^3 \times \Big(\dfrac{5}{21}\Big)^8 \\[1em] = \Big(\dfrac{5}{21}\Big)^{3+8} \\[1em] = \Big(\dfrac{5}{21}\Big)^{11}$

Hence, the answer is $\Big(\dfrac{5}{21}\Big)^{11}$

(ii) $\Big(\dfrac{-7}{3}\Big)^{11} \times \Big(\dfrac{-7}{3}\Big)^{13}$

Using the multiplication rule, we add the exponents: 11 + 13 = 24.

$\therefore \Big(\dfrac{-7}{3}\Big)^{11} \times \Big(\dfrac{-7}{3}\Big)^{13} \\[1em] = \Big(\dfrac{-7}{3}\Big)^{11+13} \\[1em] = \Big(\dfrac{-7}{3}\Big)^{24}$

Hence, the answer is $\Big(\dfrac{-7}{3}\Big)^{24}$

(iii) $\Big(\dfrac{13}{43}\Big)^7 ÷ \Big(\dfrac{13}{43}\Big)^2$

Using the division rule, we subtract the exponents: 7 - 2 = 5.

$\therefore \Big(\dfrac{13}{43}\Big)^7 ÷ \Big(\dfrac{13}{43}\Big)^2 \\[1em] = \Big(\dfrac{13}{43}\Big)^{7-2} \\[1em] = \Big(\dfrac{13}{43}\Big)^5$

Hence, the answer is $\Big(\dfrac{13}{43}\Big)^5$

(iv) $\Big(\dfrac{-16}{35}\Big)^{16} ÷ \Big(\dfrac{-16}{35}\Big)^3$

Using the division rule, we subtract the exponents: 16 - 3 = 13.

$\therefore \Big(\dfrac{-16}{35}\Big)^{16} ÷ \Big(\dfrac{-16}{35}\Big)^3 \\[1em] = \Big(\dfrac{-16}{35}\Big)^{16-3} \\[1em] = \Big(\dfrac{-16}{35}\Big)^{13}$

Hence, the answer is $\Big(\dfrac{-16}{35}\Big)^{13}$

(v) $\Big(\dfrac{-7}{15}\Big)^{12} ÷ \Big(\dfrac{-7}{15}\Big)^{15}$

Using the division rule, we subtract the exponents: 12 - 15 = -3.

$\therefore \Big(\dfrac{-7}{15}\Big)^{12} ÷ \Big(\dfrac{-7}{15}\Big)^{15} \\[1em] = \Big(\dfrac{-7}{15}\Big)^{12-15} \\[1em] = \Big(\dfrac{-7}{15}\Big)^{-3} \quad \text{[The power is negative, so we take the reciprocal]} \\[1em] = \Big(\dfrac{-15}{7}\Big)^3$

Hence, the answer is $\Big(\dfrac{-15}{7}\Big)^3$

(vi) $\Big(\dfrac{1}{24}\Big)^{13} ÷ \Big(\dfrac{1}{24}\Big)^{16}$

Using the division rule, we subtract the exponents: 13 - 16 = -3.

$\therefore \Big(\dfrac{1}{24}\Big)^{13} ÷ \Big(\dfrac{1}{24}\Big)^{16} \\[1em] = \Big(\dfrac{1}{24}\Big)^{13-16} \\[1em] = \Big(\dfrac{1}{24}\Big)^{-3} \quad \text{[The power is negative, so we take the reciprocal]} \\[1em] = 24^3$

Hence, the answer is 24³

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