Solve each equation. x^5/2 = 32

Verified step by step guidance

1. Identify the equation given: \(x^{\frac{5}{2}} = 32\).
2. To solve for \(x\), isolate \(x\) by raising both sides of the equation to the reciprocal power of \(\frac{5}{2}\), which is \(\frac{2}{5}\). This step uses the property \((a^{m})^{n} = a^{mn}\).
3. Apply the reciprocal exponent to both sides: \(\left(x^{\frac{5}{2}}\right)^{\frac{2}{5}} = 32^{\frac{2}{5}}\).
4. Simplify the left side using the exponent rule: \(x^{\left(\frac{5}{2} \times \frac{2}{5}\right)} = x^{1} = x\).
5. Now, express \$32\( as a power of 2 (since \)32 = 2^5$) to simplify \(32^{\frac{2}{5}}\) as \(\left(2^5\right)^{\frac{2}{5}} = 2^{5 \times \frac{2}{5}} = 2^2\).

Key Concepts

Exponents and Rational Exponents

Exponents indicate how many times a base is multiplied by itself. Rational exponents, like x^(m/n), represent roots and powers simultaneously, where the denominator n is the root and the numerator m is the power. For example, x^(5/2) means the square root of x raised to the 5th power.

Solving Equations with Rational Exponents

To solve equations involving rational exponents, isolate the term with the exponent and then apply inverse operations, such as raising both sides to the reciprocal power. This helps eliminate the fractional exponent and simplifies the equation to a solvable form.

Properties of Equality and Inverse Operations

Properties of equality allow performing the same operation on both sides of an equation without changing its solution. Inverse operations, like taking roots or powers, are used to undo exponents and isolate the variable, enabling the solution of the equation.