Sure, let's break down each part step by step using the given expressions [tex]\( x = 2^p \)[/tex] and [tex]\( y = 2^q \)[/tex].
### Part (i) Express [tex]\( 2^{p+q} \)[/tex] in terms of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
To express [tex]\( 2^{p+q} \)[/tex] in terms of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
1. Since [tex]\( x = 2^p \)[/tex] and [tex]\( y = 2^q \)[/tex], we can rewrite these expressions as:
[tex]\[ 2^p = x \][/tex]
[tex]\[ 2^q = y \][/tex]
2. To find [tex]\( 2^{p+q} \)[/tex], we use the property of exponents that states [tex]\( 2^{p+q} = 2^p \cdot 2^q \)[/tex].
3. Substitute [tex]\( 2^p \)[/tex] and [tex]\( 2^q \)[/tex] with [tex]\( x \)[/tex] and [tex]\( y \)[/tex] respectively:
[tex]\[ 2^{p+q} = (2^p) \cdot (2^q) = x \cdot y \][/tex]
So, [tex]\( 2^{p+q} = x \cdot y \)[/tex].
### Part (ii) Express [tex]\( 2^{2p} \)[/tex] in terms of [tex]\( x \)[/tex]
To express [tex]\( 2^{2p} \)[/tex] in terms of [tex]\( x \)[/tex]:
1. Recall that [tex]\( x = 2^p \)[/tex].
2. To find [tex]\( 2^{2p} \)[/tex], we recognize that [tex]\( 2^{2p} = (2^p)^2 \)[/tex].
3. Substitute [tex]\( 2^p \)[/tex] with [tex]\( x \)[/tex]:
[tex]\[ 2^{2p} = (2^p)^2 = x^2 \][/tex]
So, [tex]\( 2^{2p} = x^2 \)[/tex].
### Part (iii) Express [tex]\( 2^{q-1} \)[/tex] in terms of [tex]\( y \)[/tex]
To express [tex]\( 2^{q-1} \)[/tex] in terms of [tex]\( y \)[/tex]:
1. Recall that [tex]\( y = 2^q \)[/tex].
2. To find [tex]\( 2^{q-1} \)[/tex], we recognize that [tex]\( 2^{q-1} = \frac{2^q}{2} \)[/tex].
3. Substitute [tex]\( 2^q \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ 2^{q-1} = \frac{2^q}{2} = \frac{y}{2} \][/tex]
So, [tex]\( 2^{q-1} = \frac{y}{2} \)[/tex].
### Summary:
(i) [tex]\( 2^{p+q} = x \cdot y \)[/tex]
(ii) [tex]\( 2^{2p} = x^2 \)[/tex]
(iii) [tex]\( 2^{q-1} = \frac{y}{2} \)[/tex]
