Solve each equation. See Example 7. x^5/4 = 32

Verified step by step guidance

1. Identify the equation given: \(x^{\frac{5}{4}} = 32\).
2. To isolate \(x\), raise both sides of the equation to the reciprocal power of \(\frac{5}{4}\), which is \(\frac{4}{5}\). This step will help eliminate the fractional exponent on \(x\). So, write: \(\left(x^{\frac{5}{4}}\right)^{\frac{4}{5}} = 32^{\frac{4}{5}}\).
3. Simplify the left side using the property of exponents: \(\left(a^{m}\right)^{n} = a^{m \cdot n}\). This gives \(x^{\left(\frac{5}{4} \times \frac{4}{5}\right)} = x^{1} = x\).
4. Now, focus on simplifying the right side: \(32^{\frac{4}{5}}\). Express 32 as a power of 2, since \$32 = 2^{5}$, then rewrite the expression as \(\left(2^{5}\right)^{\frac{4}{5}}\).
5. Use the exponent rule again to simplify: \(\left(2^{5}\right)^{\frac{4}{5}} = 2^{5 \times \frac{4}{5}} = 2^{4}\). This will give the value of \(x\).

Key Concepts

Exponents and Powers

Exponents indicate how many times a base number is multiplied by itself. Understanding how to manipulate exponents, including fractional exponents, is essential for solving equations like x^(5/4) = 32.

Fractional Exponents

Fractional exponents represent roots and powers simultaneously; for example, x^(5/4) means the fourth root of x raised to the fifth power. Recognizing this helps in rewriting and solving the equation.

Solving Exponential Equations

Solving exponential equations often involves isolating the variable by applying inverse operations such as taking roots or raising both sides to a reciprocal power. This process is key to finding the value of x in the given equation.