Power Play

Express the following in exponential form: (i) $6 \times 6 \times 6 \times 6$ (ii) $y \times y$ (iii) $b \times b \times b \times b$ (iv) $5 \times 5 \times 7 \times 7 \times 7$ (v) $2 \times 2 \times a \times a$ (vi) $a \times a \times a \times c \times c \times c \times c \times d$

Express each of the following as a product of powers of their prime factors in exponential form. (i) $648$ (ii) $405$ (iii) $540$ (iv) $3600$

Find out the units digit in the value of $2^{224} \div 4^{32}$? [Hint: $4 = 2^2$]

Write the given number as the product of two or more powers in three different ways. The powers can be any integers. (i) $64^3$ (ii) $192^8$ (iii) $32^{-5}$

Simplify and write these in the exponential form. (i) $10^{-2} \times 10^{-5}$ (ii) $5^7 \div 5^4$ (iii) $9^{-7} \div 9^4$ (iv) $(13^{-2})^{-3}$ (v) $m^5n^{12}(mn)^9$

If $12^2 = 144$ what is (i) $(1.2)^2$ (ii) $(0.12)^2$ (iii) $(0.012)^2$ (iv) $120^2$

Circle the numbers that are the same— $2^4 \times 3^6$ $6^4 \times 3^2$ $6^{10}$ $18^2 \times 6^2$ $6^{24}$

Identify the greater number in each of the following— (i) $4^3$ or $3^4$
(ii) $2^8$ or $8^2$
(iii) $100^2$ or $2^{100}$

A dairy plans to produce 8.5 billion packets of milk in a year. They want a unique ID (identifier) code for each packet. If they choose to use the digits 0–9, how many digits should the code consist of?

64 is a square number $(8^2)$ and a cube number $(4^3)$. Are there other numbers that are both squares and cubes? Is there a way to describe such numbers in general?

The worldwide population of sheep (2024) is about $10^9$, and that of goats is also about the same. What is the total population of sheep and goats?

What was the date 1 arab/1 billion seconds ago?

Which expression describes the thickness of a sheet of paper after it is folded 10 times? The initial thickness is represented by the letter-number $v$.

Write the following expressions as a power of a power in at least two different ways: (i) $8^6$ (ii) $7^{15}$ (iii) $9^{14}$ (iv) $5^8$

The number of lotuses doubles every day in a pond. After 30 days, the pond is completely covered with lotuses. Write the number of lotuses (in exponential form) when the pond was — (i) fully covered (ii) half covered

Compute the value of $2^5 \times 5^5$.

Simplify $\frac{10^4}{5^4}$ and write it in exponential form.

Roxie has 7 dresses, 2 hats, and 3 pairs of shoes. How many different ways can Roxie dress up?

How many 5-digit passwords are possible?

Estu says, 'Next time, I will buy a lock that has 6 slots with the letters A to Z. I feel it is safer.' How many passwords are possible with such a lock?