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Power of a power with variable bases

In Power of powers, we learnt the process of how to simplify a term expressed as the power of a power.

The power of a power law

For any base number $a$a, and any numbers $m$m and $n$n as powers,

$\left(a^m\right)^n=a^{m\times n}$(am)n=am×n

That is, when simplifying a term with a power that itself has a power:

  • Keep the same base
  • Find the product of the powers


Here we will look at powers that have variable bases, and also a mix of variable and integer bases. Since "powers of powers" involve expressions with brackets, it's important to remember that everything inside the brackets is raised to the outside power.



Let's say we want to simplify the expression $\left(2x^2\right)^3$(2x2)3:

A common mistake is to only apply the outside power to the algebraic term. If we did this, we would get an answer of $2x^{2\times3}=2x^6$2x2×3=2x6, which is not correct. 

Consider the expression in expanded form: $\left(2x^2\right)^3=2x^2\times2x^2\times2x^2$(2x2)3=2x2×2x2×2x2

$\left(2x^2\right)^3$(2x2)3 $=$= $2x^2\times2x^2\times2x^2$2x2×2x2×2x2
  $=$= $\left(2\times2\times2\right)\times\left(x^2\times x^2\times x^2\right)$(2×2×2)×(x2×x2×x2)
  $=$= $2^3\times\left(x^2\right)^3$23×(x2)3
  $=$= $8x^6$8x6

You can see that not only is $x^2$x2 multiplied $3$3 times, $\left(x^2\right)^3$(x2)3, but $2$2 is also multiplied $3$3 times, $2^3$23.

So we need to raise $2$2 to the power of $3$3 as well as $x^2$x2 to the power of $3$3.



Beware of the signs when raising negative numbers to a power, as a negative number raised to an odd power has a negative result. For example:




Question 1

Simplify the following, giving your answer with a positive exponent: $\left(w^3\right)^4$(w3)4

Question 2

Simplify, and evaluate where possible, the following expression:


Question 3

Simplify the following expression: $\left(3a^3b^2c^3\right)^3$(3a3b2c3)3.



Extend the multiplication rule to derive and understand the power of a power rule, and apply it to simplify expressions involving one and two variables with positive exponents

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