Question

Hey guys, need help here

Answer 1

Assuming
`a` is a non-negative real number...

Raise both sides to the power 2 :

`a^(3/4) = 8 \implies \left(a^(3/4)\right)^2 = 8^2 \implies a^(3/2) = 64`

which follows from the exponent product property,
`(x^y)^z=x^(yz)`. In particular, 3/4 × 2 = 6/4 = 3/2.

Raise both sides to the power 1/3 :

`\left(a^(3/2)\right)^(1/3) = 64^(1/3) \implies a^(1/2) = 4`

where we use the same property as before, and 4³ = 64. This time, 3/2 × 1/3 = 3/6 = 1/2.

Take the reciprocal of both sides. This negates the exponent, so we end up with

`\frac1{a^(1/2)} = \frac14 \implies \boxed{a^(-1/2) = \frac14}`

Of course, you could also solve for
`a` immediately by raising both sides of the original equation to the power 4/3. We have 4/3 × 3/4 = 12/12 = 1, so

`a^(3/4) = 8 \implies \left(a^(3/4)\right)^(4/3) = 8^(4/3) \implies a = 8^(4/3)`

Now 2³ = 8, so
`8^(4/3) = \left(8^(1/3)\right)^4 = 2^4 = 16`, and since 4² = 16, it follows that

`a = 16 \implies a^(1/2) = √(16) = 4 \implies a^(-1/2) = \frac14`