Solve p/56=frac{sqrt{784}*sqrt{49}}{4p^2}

$$\frac { p } { 56 } = \frac { \sqrt { 784 } \times \sqrt { 49 } } { 4 p ^ { 2 } }$$

Solve for p

$$p=14$$

Solution Steps

Variable p cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 56p^{2}, the least common multiple of 56,4p^{2}.

$$p^{2}p=14\sqrt{784}\sqrt{49}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$p^{3}=14\sqrt{784}\sqrt{49}$$

Factor 784=49\times 16. Rewrite the square root of the product \sqrt{49\times 16} as the product of square roots \sqrt{49}\sqrt{16}.

$$p^{3}=14\sqrt{49}\sqrt{16}\sqrt{49}$$

Multiply \sqrt{49} and \sqrt{49} to get 49.

$$p^{3}=14\times 49\sqrt{16}$$

Multiply 14 and 49 to get 686.

$$p^{3}=686\sqrt{16}$$

Calculate the square root of 16 and get 4.

$$p^{3}=686\times 4$$

Multiply 686 and 4 to get 2744.

$$p^{3}=2744$$

Subtract 2744 from both sides.

$$p^{3}-2744=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -2744 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±2744,±1372,±686,±392,±343,±196,±98,±56,±49,±28,±14,±8,±7,±4,±2,±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$p=14$$

By Factor theorem, p-k is a factor of the polynomial for each root k. Divide p^{3}-2744 by p-14 to get p^{2}+14p+196. Solve the equation where the result equals to 0.

$$p^{2}+14p+196=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 14 for b, and 196 for c in the quadratic formula.

$$p=\frac{-14±\sqrt{14^{2}-4\times 1\times 196}}{2}$$

Do the calculations.

$$p=\frac{-14±\sqrt{-588}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions.

$$p\in \emptyset$$

List all found solutions.

$$p=14$$