Properties of powers

In the school games room, board games are organised on shelves. If we have an expression in power form such as $7^{3}$, state which number is the base and which is the exponent. Briefly explain what operation you would need to perform to calculate its value (without needing to give the final result).

A well-known social network displays user profile photos in a square grid. If Lucía's photo feed is $5$ photos wide and $5$ photos high, how many photos does she have in total? Express the result first as a multiplication of two equal factors, then as a power, and finally calculate the total value.

Calculate the following special powers mentally. Remember the rules for exponents $0$ and $1$:

1. a) $15^{1} =$
2. b) $8^{0} =$
3. c) $1^{25} =$
4. d) $10^{1} =$
5. e) $2024^{0} =$

Apply the rule for the product of powers with the same base to write the following expressions as a single power (it is not necessary to calculate the final value):

1. a) $3^{2} \cdot 3^{4} =$
2. b) $5^{3} \cdot 5^{3} =$
3. c) $a^{2} \cdot a^{5} =$
4. d) $10^{2} \cdot 10 \cdot 10^{3} =$ (Hint: when a number has no exponent, it is as if it has a $1$).

Use the rule for the quotient of powers with the same base to simplify these operations into a single power:

1. a) $7^{5} : 7^{2} =$
2. b) $10^{8} : 10^{6} =$
3. c) $\frac{4^{9}}{4^{7}} =$
4. d) $m^{10} : m^{9} =$

A courier company organises its packages in large cubic containers. If a container fits $6$ boxes along its length, $6$ boxes along its width and $6$ boxes along its height:

1. a) Express the total number of boxes in a container as a power and calculate the result.
2. b) If the company has $6^2$ identical containers stored, how many boxes do they have in total? Express the result as a single power.

Write as a single power using the power of a power rule:

1. a) $(2^{3})^{2} =$
2. b) $(5^{4})^{3} =$
3. c) $(10^{2})^{5} =$
4. d) $(x^{3})^{4} =$

Use the laws of indices to simplify the following expression and leave it as a single power with base $2$:

Remember to show the steps followed: first work out the inside of the brackets, then apply power of a power, and finally work out the quotient.

Calculate the following expressions in two different ways to check that the laws of indices hold true:

• Method A: By calculating the inside of the brackets first and then the power.

• Method B: By applying the rule for the power of a product or a quotient.

1. a) $(2 \cdot 5)^3$
2. b) $(12 : 3)^2$
3. c) $(2^2 \cdot 3)^2$

Find the value of the letter $x$ in each of the following equations using the laws of indices:

1. a) $5^x \cdot 5^4 = 5^9$
2. b) $(3^2)^x = 3^6$
3. c) $10^7 : 10^x = 10^3$
4. d) $x^3 = 64$
5. e) $x^0 = 1$ (State which values $x$ cannot take in this case).

Solve the following mixed operations, applying the order of operations correctly and the laws of indices where possible:

1. a) $3^3 - \sqrt{49} + 2^2 \cdot (5 - 3)^2$
2. b) $(\sqrt{100} - 2^3)^2 + 4^0 \cdot 5$
3. c) $2^5 : 2^3 + \sqrt{81} \cdot 3 - 1^{15}$