Q.Simplify $\dfrac{x^5\cdot x^{-2}}{x^{1/2}}$, expressing the answer with a single positive or fractional index.

Combine the numerator using the product law first, then apply the quotient law against the denominator.

Step 1 — combine the numerator (product law):

Step 2 — apply the quotient law against the denominator:

Check (independent recomputation): combine all three exponents directly in one sum, since every operation here is on the same base $x$: numerator exponents add ($5+(-2)=3$), then the denominator's exponent is subtracte …

Forgetting the negative sign on the $-2$ exponent when combining the numerator (wrongly getting $x^{5+2}=x^7$ instead of $x^3$), or subtracting $1/2$ incorrectly at the f …