Question

4.

In this question you must show all steps of your working.
Solutions relying on calculator technology are not acceptable.
(i) Given
£4
use the laws of indices to write a in terms of b.
(ii) Solve the equation
= 32√2
3.x=x√√2 +
giving your answer as a simplified surd.
+ 14
€
(3)
Leave
blank

Answer 1

`\textsf{a)} \quad a=(6b+11)/(4)`

`\textsf{b)} \quad x=6+2√(2)`

Explanation:

Part (a)

Given equation:

`(4^a)/(2^(3b))=32√(2)`

Rewrite 4 as 2² and 32 as 2⁵ and √2 as
`\sf 2^{(1)/(2)}` :

`\implies ((2^2)^a)/(2^(3b))=2^5 \cdot 2^{(1)/(2)}`

`\textsf{Apply exponent rule} \quad (a^b)^c=a^(bc):`

`\implies (2^(2a))/(2^(3b))=2^5 \cdot 2^{(1)/(2)}`

`\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^(b+c):`

`\implies (2^(2a))/(2^(3b))=2^{(11)/(2)}`

`\textsf{Apply exponent rule} \quad (a^b)/(a^c)=a^(b-c):`

`\implies 2^(2a-3b)=2^{(11)/(2)}`

`\textsf{Apply exponent rule} \quad a^(f(x))=a^(g(x)) \implies f(x)=g(x):`

`\implies 2a-3b=(11)/(2)`

Add 3b to both sides:

`\implies 2a=3b+(11)/(2)`

Divide both sides by 2:

`\implies a=(6b+11)/(4)`

Part (b)

Given equation:

`3x=x√(2)+14`

Subtract x√2 from both sides:

`\implies 3x-x√(2)=14`

Factor out x:

`\implies x(3-√(2))=14`

Divide both sides by (3 - √2):

`\implies x=(14)/(3-√(2))`

Multiply the numerator and the denominator by the conjugate of the denominator (3 - √2):

`\implies x=(14(3+√(2)))/((3-√(2))(3+√(2)))`

`\implies x=(42+14√(2))/(9+3√(2)-3√(2)-2)`

`\implies x=(42+14√(2))/(7)`

`\implies x=(42)/(7)+(14√(2))/(7)}`

`\implies x=6+2√(2)`