Solve each equation. See Examples 4–6. x^5/2 = 32

Verified step by step guidance

1. Recognize that the equation is in the form \(x^{\frac{5}{2}} = 32\).
2. To isolate \(x\), raise both sides of the equation to the reciprocal of \(\frac{5}{2}\), which is \(\frac{2}{5}\). This gives \((x^{\frac{5}{2}})^{\frac{2}{5}} = 32^{\frac{2}{5}}\).
3. Simplify the left side of the equation using the property \((a^m)^n = a^{m \cdot n}\), which results in \(x^{1} = x\).
4. Now, you have \(x = 32^{\frac{2}{5}}\).
5. To find \(32^{\frac{2}{5}}\), first find the fifth root of 32, which is \(32^{\frac{1}{5}}\), and then square the result.

Key Concepts

Exponents and Radicals

Exponents represent repeated multiplication of a base number, while radicals are the inverse operation, indicating the root of a number. In the equation x^(5/2) = 32, the exponent 5/2 can be interpreted as both a power (x^5) and a root (the square root of x). Understanding how to manipulate these forms is essential for solving equations involving exponents.

Isolating the Variable

Isolating the variable is a fundamental algebraic technique used to solve equations. This involves rearranging the equation to get the variable (in this case, x) on one side and all other terms on the opposite side. For the equation x^(5/2) = 32, this means applying inverse operations to both sides to find the value of x.

Inverse Operations

Inverse operations are pairs of mathematical operations that undo each other, such as addition and subtraction or multiplication and division. In the context of exponents, taking a root is the inverse of raising to a power. To solve x^(5/2) = 32, one would raise both sides to the reciprocal of 5/2, which is 2/5, to isolate x.