Powers

Express the following products as a power and state the base and the exponent in each case:

1. a) $5 \times 5 \times 5 \times 5$

1. b) $12 \times 12 \times 12$

1. c) $2 \times 2 \times 2 \times 2 \times 2 \times 2$

1. d) $10 \times 10 \times 10 \times 10 \times 10$

Work out the value of the following basic powers:

1. a) $3^{4}$

1. b) $2^{5}$

1. c) $7^{2}$

1. d) $11^{0}$

1. e) $1^{25}$

Write the polynomial expansion of the following numbers using powers of base $10$:

1. a) $4\,527$

1. b) $30\,816$

1. c) $504$

Express the following data using standard form:

a) The mean distance from the Earth to the Moon is about $384\,400$ kilometres.

b) The approximate world population is $8\,000\,000\,000$ people.

c) One light-year is approximately equal to $9\,461\,000\,000\,000$ kilometres.

Apply the laws of indices to write the result as a single power:

1. a) $3^{4} \times 3^{2} \times 3^{5}$

1. b) $a^{3} \times a^{4}$

1. c) $10^{2} \times 10 \times 10^{3}$

1. d) $2^{6} \times 2^{6}$

Simplify to a single power using the division law of indices for powers with the same base:

1. a) $5^{8} : 5^{3}$

1. b) $x^{10} : x^{7}$

1. c) $10^{6} : 10^{5}$

1. d) $\dfrac{2^{9}}{2^{4}}$

Apply the laws of indices to simplify the following expressions to a single power:

a) $2^5 \times 2^3 \times 2 =$

b) $(3^4)^3 : 3^7 =$

c) $\frac{5^8 \times 5^2}{5^7} =$

Simplify the following expressions, leaving the result as a single power:

1. a) $(4^{3})^{2}$

1. b) $(10^{5})^{4}$

1. c) $(x^{2})^{8}$

In a logistics warehouse, boxes are organised into cubic shelving units. A storage block consists of 4 shelving units in width, 4 in length, and 4 in height. If each shelving unit contains 4 boxes:

a) Express the total number of boxes as a single power.

b) If each box contains $2^3$ packs of paper, how many packs of paper are there in total in the block? Express the result as a power of base 2.

Work out the value of the following combined expressions respecting the order of operations:

a) $2^4 - 3^2 + \sqrt{25} \times 2 =$

b) $(5-2)^3 + 10^2 : (2^2 + 1) =$

c) $2 \times (3^2 - 4) + \sqrt{64} : 2^3 =$

A square mosaic made of square tiles is to be designed. It is known that the number of tiles used is the solution to the following operation:

$$ \sqrt{10^2 - 8^2} \times (3^3 - 5^2)^2 - 2^3 $$

1. a) How many tiles will the mosaic have in total?
2. b) How many tiles will there be along each side of the square mosaic?

A video by a streamer has gone viral on a social network. During the first few hours, the number of people sharing the video triples every hour. If 3 people shared it at the start:

a) Write as a power the number of people who will have shared the video after 5 hours.

b) Work out the value of this power to find the total number of people at that time.

Find the value of the unknown $n$ in the following equations, applying the laws of indices:

1. a) $2^n \times 2^3 \times 2 = 1024$
2. b) $5^6 : 5^n = 125$
3. c) $(3^2)^n \times 9 = 3^{10}$

A natural number $X$ has the following prime factorization: $X = 2^a \times 3^2 \times 5^b$.

It is known that:

• The greatest common divisor of $X$ and $180$ is $2^2 \times 3^2 \times 5^0$.

• The least common multiple of $X$ and $180$ is $2^3 \times 3^2 \times 5^1$.

Find the values of the exponents $a$ and $b$ and write the number $X$ in decimal notation. (Hint: $180 = 2^2 \times 3^2 \times 5^1$).