Question

Write \(8^{\,2x+3}\) in the form \(2^{\,y}\) and express the relation between \(x\) and \(y\).

Hint:

To convert between exponential forms, first rewrite all numbers with the same base, then equate exponents.

Answer 1

Step 1 (Rewrite the base 8 using base 2).
Since \(8 = 2^3\), we can rewrite the given power as \[ 8^{\,2x+3} = (2^3)^{\,2x+3}. \] Step 2 (Use the power-of-a-power rule).
The identity \((a^m)^n = a^{mn}\) gives \[ (2^3)^{\,2x+3} = 2^{\,3(2x+3)}. \] Step 3 (Match with \(2^{\,y}\)).
If \(8^{\,2x+3} = 2^{\,y}\), then the exponents must be equal: \[ y = 3(2x+3) = 6x + 9. \] Step 4 (Tiny check).
Take \(x=1\) for a check: \(8^{2(1)+3}=8^{5}=(2^3)^5=2^{15}\). Our relation gives \(y=6(1)+9=15\) — consistent.
\[ {y = 6x + 9} \]