Question

6^3/2 + 6^1/2= 7 square root of 6

We know the answer but don’t understand how they got it.

Can someone explain?

Answer 1

`6^{(3)/(2)}+6^{(1)/(2)}=7√(6)`

Explanation:

Given expression:

`6^{(3)/(2)}+6^{(1)/(2)}`

Begin by rewriting the exponent of the first term as 1 + ¹/₂:

`\implies 6^{1+(1)/(2)}+6^{(1)/(2)}`

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

`\implies 6^(1) \cdot 6^{(1)/(2)}+6^{(1)/(2)}`

`\textsf{Apply the exponent rule:} \quad a^1=a`

`\implies 6 \cdot 6^{(1)/(2)}+6^{(1)/(2)}`

`\textsf{Rewrite\;$6^{(1)/(2)}$\;as\;$1 \cdot 6^{(1)/(2)}$:}`

`\implies 6 \cdot 6^{(1)/(2)}+1\cdot6^{(1)/(2)}`

`\textsf{Factor out}\;6^{(1)/(2)}:`

`\implies \left(6+1\right)6^{(1)/(2)}`

Simplify:

`\implies 7 \cdot 6^{(1)/(2)}`

`\textsf{Apply the exponent rule:} \quad a^{(1)/(2)}=√(a), \;\;a\geq0`

`\implies 7 √(6)`