HClO4+P4O10=H3PO4+Cl2O7 how do you balance this?

Assign coefficient 1 to the species with the largest number of elements and atoms. H₃PO₄ contains three elements. Each of its molecule contains eight atoms, that's two more than the six atoms in a HClO₄ molecule. Start by assigning H₃PO₄ a coefficient of 1.

? HClO₄ + ? P₄O₁₀ → 1 H₃PO₄ + ? Cl₂O₇

There are now three H atoms, one P atom on the product side. H₃PO₄ is the only product that contains H and P atoms. As a result, there should be the same number of H and P atoms on the reactant side.

Among all reactants, only HClO₄ contains H atoms. Each HClO₄ molecule contains one H atom. Three H atoms correspond to three HClO₄ molecule.
Among all reactants, only P₄O₁₀ contains P atoms. Each P₄O₁₀ molecule contains four P atoms. One P atom corresponds to 1/4 of a P₄O₁₀ molecule.

Thus

3 HClO₄ + 1/4 P₄O₁₀ → 1 H₃PO₄ + ? Cl₂O₇

There are three Cl atoms in three HClO₄ molecules. HClO₄ is the only species that contains Cl among all reactants. There are three Cl atoms on the reactant side and shall be the same number of Cl atoms on the product side.

Cl₂O₇ is the only molecule that contains Cl among the products. Each Cl₂O₇ molecule contains two Cl atoms. Three Cl atoms will correspond to 3/2 Cl₂O₇ molecules.

3 HClO₄ + 1/4 P₄O₁₀ → 1 H₃PO₄ + 3/2 Cl₂O₇

Multiply both sides by the least common multiple of the denominators to eliminate the fraction. The least common multiple in this case is four.

12 HClO₄ + 1 P₄O₁₀ → 4 H₃PO₄ + 6 Cl₂O₇