Powers

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

1. a) $5 \cdot 5 \cdot 5 \cdot 5$

1. b) $12 \cdot 12 \cdot 12$

1. c) $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$

1. d) $10 \cdot 10 \cdot 10 \cdot 10 \cdot 10$

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) $1^{25}$

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

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

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

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

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

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) $2^5 \cdot 2^3 \cdot 2 =$

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

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

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

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

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

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