Solve each equation. x^3/2 = 125

Verified step by step guidance

1. Identify the equation given: \(x^{\frac{3}{2}} = 125\).
2. Recall that the exponent \(\frac{3}{2}\) means the square root of \(x\) raised to the third power, or equivalently, \((x^{\frac{1}{2}})^3\).
3. To isolate \(x\), raise both sides of the equation to the reciprocal power of \(\frac{3}{2}\), which is \(\frac{2}{3}\), so you get \(\left(x^{\frac{3}{2}}\right)^{\frac{2}{3}} = 125^{\frac{2}{3}}\).
4. Simplify the left side using the property of exponents: \(\left(x^{a}\right)^{b} = x^{a \cdot b}\), so \(x^{\frac{3}{2} \cdot \frac{2}{3}} = x^1 = x\).
5. Evaluate the right side by first finding the cube root of 125, then squaring the result: \(125^{\frac{1}{3}}\) and then square it to get \(125^{\frac{2}{3}}\).

Key Concepts

Rational Exponents

Rational exponents represent roots and powers combined. For example, x^(3/2) means the square root of x cubed, or (√x)^3. Understanding how to manipulate and interpret these exponents is essential for solving equations involving fractional powers.

Isolating the Variable

To solve equations like x^(3/2) = 125, you first isolate the variable expression. This often involves applying inverse operations such as raising both sides to the reciprocal power to undo the rational exponent and solve for x.

Properties of Exponents

Properties of exponents, such as (a^m)^n = a^(mn), help simplify expressions and solve equations. Applying these rules correctly allows you to manipulate the equation and find the value of the variable efficiently.