Indices
India
Class IX
Power of a power with integer or variable bases and negative powers
Lesson
We've already learnt about the power of a power law, which states:
$\left(a^x\right)^y=a^{xy}$(ax)y=axy
We can still use this rule for questions with negative indices. We just need to apply the same rules that we learnt when we learnt to multiply negative numbers.
Remember!
- Multiplying two negative numbers gives a positive answer, e.g. $-5\times\left(-3\right)=15$−5×(−3)=15
- Multiplying a positive and a negative number gives a negative answer, e.g. $6\times\left(-9\right)=-54$6×(−9)=−54
Examples
Question 1
Simplify the following into the form $a^b$ab:
$\left(6^7\right)^{-3}$(67)−3
Question 2
Simplify, expressing in positive index form.
$\left(4y^4\right)^{-4}$(4y4)−4
Question 3
Simplify the following, giving your answer with positive indices: $\left(\frac{a^3}{b^3}\right)^{-5}$(a3b3)−5