In this chapter we will discuss the power of power rule for exponents with solved examples.
What is the power of power rule ?
According to this rule, when exponent is raised raised by another power then the expression can be simplified by multiplication of powers.
Consider the exponent \mathtt{\left( a^{b}\right)^{c}} .
Here the base is “a”.
The number “a” is raised to power “b”, which is again raised to power “c”.
To simplify the expression, just multiply the powers.
\mathtt{\left( a^{b}\right)^{c} \ =\ a^{b\ \times \ c}}
I hope you understood the above concept. Let us solve some examples for further clarity.
Power rule for exponents – Solved examples
Example 01
Simplify \mathtt{\left( 2^{3}\right)^{5}}
Solution
Here the base is number 2.
The number 2 is raised to the power 3 which is again raised to the power 5.
To simplify the expression, just multiply both the powers.
\mathtt{\Longrightarrow \ \left( 2^{3}\right)^{5} \ }\\\ \\ \mathtt{\Longrightarrow \ 2^{3\ \times \ 5}}\\\ \\ \mathtt{\Longrightarrow \ 2^{15}}
Hence, \mathtt{\ 2^{15}} is the simplified form of given expression.
Example 02
Simplify the expression \mathtt{\left( 18^{-6}\right)^{3} \ }
Solution
Here the base number 18 is raised to power -6 and then raised to another power 3.
Note that the power rule of exponents is also applicable for negative powers.
To find the solution, simply multiply the exponents.
\mathtt{\Longrightarrow \ \left( 18^{-6}\right)^{3} \ }\\\ \\ \mathtt{\Longrightarrow \ 18^{-6\ \times \ 3}}\\\ \\ \mathtt{\Longrightarrow \ 18^{-18}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{18^{18}}}
Example 03
Simplify \mathtt{\left(\left( 75^{2}\right)^{7}\right)^{4} \ }
Solution
This is an example of power of power of power.
Note that for more than two powers also, the above rule is applicable.
So in order to simplify the expression, just multiply the powers.
\mathtt{\Longrightarrow \ \left(\left( 75^{2}\right)^{7}\right)^{4} \ }\\\ \\ \mathtt{\Longrightarrow \ 75^{2\ \times \ 7\ \times \ 4}}\\\ \\ \mathtt{\Longrightarrow \ 75^{56}}
Hence, \mathtt{75^{56}} is the simplified form of given expression.
Example 04
Simplify \mathtt{\left( 11^{6}\right)^{-9} \ }
Solution
For simplification, simply multiply the powers.
\mathtt{\Longrightarrow \ \left( 11^{6}\right)^{-9} \ }\\\ \\ \mathtt{\Longrightarrow \ 11^{6\ \times -9}}\\\ \\ \mathtt{\Longrightarrow \ 11^{-54}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{11^{54}}}
Example 05
Simplify \mathtt{\left( -13^{11}\right)^{2} \ }
Solution
\mathtt{\Longrightarrow \ \left( -13^{11}\right)^{2} \ }\\\ \\ \mathtt{\Longrightarrow \ -13^{11\ \times \ 2}}\\\ \\ \mathtt{\Longrightarrow \ -13^{22}}
Example 06
Simplify \mathtt{\left( 6\ x^{9}\right)^{5} \ }
Solution
The above expression contain base 6 & x in multiplication form.
Base 6 is raised to single power 9.
Base x is first raised to power 9 and then power 5.
Here we have to simplify both base 6 & 9 separately.
\mathtt{\Longrightarrow \ \left( 6\ x^{9}\right)^{5} \ }\\\ \\ \mathtt{\Longrightarrow \ 6^{5} \times \ \left( x^{9}\right)^{5}}\\\ \\ \mathtt{\Longrightarrow \ 6^{5} \times \ x^{9\ \times \ 5}}\\\ \\ \mathtt{\Longrightarrow \ 6^{5} \times x^{45}}
Hence, the above expression is simplified to \mathtt{\ 6^{5} .x^{45}}
Note: When two or more bases are present in multiplication or division form, the separation of powers can be done directly.
Example 07
Simplify \mathtt{\left(\frac{11}{x^{3}}\right)^{6}}
Solution
Here the two base 11 and x are present in division form.
We know that when two or more bases are present in multiplication or division form, the separation of power can be done directly.
\mathtt{\Longrightarrow \ \left(\frac{11}{x^{3}}\right)^{6} \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{11^{6}}{\left( x^{3}\right)^{6}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{11^{6}}{\left( x^{3\times 6}\right)}}\\\ \\ \mathtt{\Longrightarrow \ \frac{11^{6}}{\left( x^{18}\right)}}
Example 08
Simplify \mathtt{\ \left(\frac{x^{2}}{y^{5} z}\right)^{9}}
Solution
In the given expression, there are three bases x, y and z present in multiplication and division form.
We know that in multiplication and division of bases, the power can be separated directly.
Simplifying the given expression;
\mathtt{\Longrightarrow \ \left(\frac{x^{2}}{y^{5} z}\right)^{9} \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{\left( x^{2}\right)^{9}}{\left( y^{5}\right)^{9} .z^{9}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{x^{2\ \times \ 9}}{\left( y^{5\times 9}\right) z^{9}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{x^{18}}{\left( y^{45}\right) \ z{^{9}}}}