Multiplication law with integer and variable bases

In Multiplying Powers, we learnt about the multiplication law. The multiplication law states:

$a^x\times a^y=a^{x+y}$ax×ay=ax+y

Now we are going to to build on this knowledge to simplify expressions involving more than one pronumeral.

Of course this multiplication law can be applied any time we are multiplying terms with the same base. If we have an expression which consists of more than one base, we need to apply the multiplication law separately to each set of like bases.

In an expression such as $x^2y\times xy^2$x2y×xy2, we must simplify the powers of $x^2\times x$x2×x separately to $y\times y^2$y×y2.

It may be useful to split up and rearrange the multiplication to make it obvious which terms have like bases and can therefore be simplified. For example, you may like to rewrite $6ab\times a^2$6ab×a2 as $6\times b\times a\times a^2$6×b×a×a2 to identify that only the powers of $a\times a^2$a×a2 can be added.

 

Examples

Question 1

Simplify: $8x^3y^2\times4x^5y^7$8x3y2×4x5y7

Think: We will multiply the coefficients, then evaluate the power of $x$x separately from the power of $y$y.

$8x^3y^2\times4x^5y^7$8x3y2×4x5y7	$=$=	$32x^3y^2\times x^5y^7$32x3y2×x5y7
	$=$=	$32x^8y^2\times y^7$32x8y2×y7
	$=$=	$32x^8y^9$32x8y9

 

Question 2

 

Question 3

Question 4