Equations with Surds and Exponents

Equations with Surds and Exponents

Equations with exponents or roots need reversible operations and a check. Make powers have the same base where possible. When squaring both sides to remove a root, substitute answers into the original equation because squaring can create an extraneous solution. Keep domain restrictions visible: an even root needs a non-negative radicand in real-number work.

Worked reasoning

Solve $2^{x+1}=16$. Since $16=2^4$, $x+1=4$, so $x=3$. Check: $2^4=16$.

Exam method

1. Isolate a power or root. 2. Use matching bases or an inverse operation. 3. Substitute the answer into the original equation.

Check for meaning, not only arithmetic

Every algebraic transformation must preserve the original condition. A solution to $\sqrt{x-2}=x-4$ must make both sides defined and non-negative before it can work. After isolating and squaring, substitute candidate values back into the root equation, not merely the squared version. For exponential equations, estimate location using powers: if $2^5=32$ and $2^6=64$, a positive answer associated with 50 must lie between 5 and 6.

One-minute retrieval: Equations with Surds and Exponents

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.