1. Simplify: (a) $m' \times m^n$ (b) Simplify: $(5mp)^{y}$ (c) Simplify: \( \frac{32q^2r^{4}}{4q^{r}} \) - Edexcel - GCSE Maths - Question 2 - 2018 - Paper 2

For the expression $(5mp)^{y}$, we use the power of a product rule, which states that $(ab)^{n} = a^{n} b^{n}$. Therefore, we have:

$(5mp)^{y} = 5^{y} m^{y} p^{y}$

This means the simplified expression is $5^{y} m^{y} p^{y}$.

To simplify the fraction ( \frac{32q^2r^{4}}{4q^{r}} ), we first simplify the coefficients and then the variable terms:

1. Coefficients: ( \frac{32}{4} = 8 )
2. Variable terms: Since we have ( q^{2} \div q^{r} ), we apply the rule ( a^{m} \div a^{n} = a^{m-n} ):
   • If ( r = 0 ): ( q^{2-0} = q^{2} )
   • If ( r = 1 ): ( q^{2-1} = q^{1} )
   • If ( r = 2 ): ( q^{2-2} = q^{0} = 1 )
   • Continue as needed depending on the context.
3. So, for the simplified expression, it can be stated as:

$8q^{2-r}r^{4}$

This gives us the final simplified form as ( 8q^{2-r} r^{4} ).