How do you simplify the expression \[{\left( {{z^5}} \right)^3}\] ?

Hint: Here we have a simple algebraic expression. We can simplify this using the laws of indices. Without knowing the laws of indices it is impossible to simplify any mathematical equation or expression. To solve this we use the third law of indices (brackets). That is \[ \Rightarrow {\left( {{x^m}} \right)^n} = {x^{m \times n}}\]

Complete step-by-step answer:
As we know the third law: brackets: If a term with a power is itself raised to a power then the powers are multiplied together. That is \[ \Rightarrow {\left( {{x^m}} \right)^n} = {x^{m \times n}}\]
Given, \[{\left( {{z^5}} \right)^3}\]
On comparing we can say that \[m = 5,n = 3\] .
Applying we have,
 \[ \Rightarrow {\left( {{z^5}} \right)^3} = {\left( z \right)^{5 \times 3}}\]
 \[ \Rightarrow {\left( {{z^5}} \right)^3} = {\left( z \right)^{15}}\] . This is the simplified form.
Here ‘z’ can be variable (unknown value) or any constant.

Note: We have several laws of indices.
 \[ \bullet \] The first law: multiplication: if the two terms have the same base and are to be multiplied together their indices are added. That is \[ \Rightarrow {x^m} \times {x^n} = {x^{m + n}}\]
 \[ \bullet \] The second law: division: If the two terms have the same base and are to be divided their indices are subtracted. That is \[ \Rightarrow \dfrac{{{x^m}}}{{{x^n}}} = {x^{m - n}}\]
 \[ \bullet \] The third law: brackets: If a term with a power is itself raised to a power then the powers are multiplied together. That is \[ \Rightarrow {\left( {{x^m}} \right)^n} = {x^{m \times n}}\]
 \[ \bullet \] As we have the second law of indices which helps to explain why anything to the power of zero is equal to one. \[ \Rightarrow {x^0} = 1\]
 \[ \bullet \] Negative power \[ \Rightarrow {x^{ - n}} = \dfrac{1}{{{x^m}}}\]
 \[ \bullet \] The fractional power \[ \Rightarrow {x^{\dfrac{m}{n}}} = \left( {\sqrt[n] {m}} \right)\]