Sure, let's break this down step by step:
### Given:
[tex]\[ x = 2^p \][/tex]
[tex]\[ y = 2^q \][/tex]
We need to express the following in terms of [tex]\( x \)[/tex] and/or [tex]\( y \)[/tex]:
### (i) [tex]\( 2^{p+q} \)[/tex]
Using the properties of exponents, we know that [tex]\( 2^{p+q} \)[/tex] can be rewritten as:
[tex]\[ 2^{p+q} = 2^p \cdot 2^q \][/tex]
Since [tex]\( 2^p = x \)[/tex] and [tex]\( 2^q = y \)[/tex], we substitute these into the equation:
[tex]\[ 2^{p+q} = x \cdot y \][/tex]
### (ii) [tex]\( 2^{2p} \)[/tex]
Using the properties of exponents, we know that [tex]\( 2^{2p} \)[/tex] can be rewritten as:
[tex]\[ 2^{2p} = (2^p)^2 \][/tex]
Since [tex]\( 2^p = x \)[/tex], we substitute this into the equation:
[tex]\[ 2^{2p} = x^2 \][/tex]
### (iii) [tex]\( 2^{4-1} \)[/tex]
First, we simplify the exponent:
[tex]\[ 2^{4-1} = 2^3 \][/tex]
Now, we evaluate the expression:
[tex]\[ 2^3 = 8 \][/tex]
In summary, the expressions we sought are:
(i) [tex]\( 2^{p+q} = x \cdot y \)[/tex]
(ii) [tex]\( 2^{2p} = x^2 \)[/tex]
(iii) [tex]\( 2^{4-1} = 8 \)[/tex]
