GCSE(F), GCSE(H),

The *Laws of Indices*, which apply to numbers, are also applied to algebra. Consider *y*^{3} x *y*^{2}.

The term *y*^{3} can be expanded to *y* x *y* x *y*. The other term, *y*^{2}, can be expanded to *y* x *y*. Therefore *y*^{3} x *y*^{2} = *y* x *y* x *y* x *y* x *y*, which is equal to *y*^{5}.

This can be written as *y*^{a} x *y*^{b} = *y*^{(a + b)}. When multiplying the same variable, add the powers.

Note that the variable letter must be the same for both the terms being multiplied.

Division works in a similar way. The rule is written as *y*^{a} ÷ *y*^{b} = *y*^{(a - b)}. When dividing the same variable, subtract the powers. The example *e*^{6} ÷ *e*^{3} can be shown as:

`frac(e^6)(e^2)`

= `frac(e times e times e times e times e times e)(e times e)`

= `frac(e times e times e times e)(1)`

= *e*^{4}

The division rule leads to the fact that *e*^{0} = 1. This can be shown by *e*^{2} ÷ *e*^{2} = *e*^{2 - 2} = *e*^{0}.

1. Simplify *x*^{3} x 4*x*^{3}.

Answer: 4*x*^{6}

*x*^{3} x 4*x*^{3}
= 4 x *x*^{(3 + 3)}
= 4*x*^{6}

2. Simplify 4*a*^{5}*b* ÷ 2*a*^{3}

Answer: 2*a*^{2}*b*

Calculate the numbers: 4 ÷ 2 = 2.

Calculate the variable *a*: *a*^{5} ÷ *a*^{3} = *a*^{(5-3)} = *a*^{2}

The variable *b* is not divided.

Put it all together: 2*a*^{2}*b*

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