Solve 4/a=a^2/16

$$\frac { 4 } { a } = \frac { a ^ { 2 } } { 16 }$$

Solve for a

$$a=4$$

Solution Steps

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 16a, the least common multiple of a,16.

$$16\times 4=aa^{2}$$

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

$$16\times 4=a^{3}$$

Multiply 16 and 4 to get 64.

$$64=a^{3}$$

Swap sides so that all variable terms are on the left hand side.

$$a^{3}=64$$

Subtract 64 from both sides.

$$a^{3}-64=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -64 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±64,±32,±16,±8,±4,±2,±1$$

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.

$$a=4$$

By Factor theorem, a-k is a factor of the polynomial for each root k. Divide a^{3}-64 by a-4 to get a^{2}+4a+16. Solve the equation where the result equals to 0.

$$a^{2}+4a+16=0$$

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, 4 for b, and 16 for c in the quadratic formula.

$$a=\frac{-4±\sqrt{4^{2}-4\times 1\times 16}}{2}$$

Do the calculations.

$$a=\frac{-4±\sqrt{-48}}{2}$$

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

$$a\in \emptyset$$

List all found solutions.

$$a=4$$