Solve each equation. See Examples 4–6. 1/27 = x^-3

Verified step by step guidance

1. Start with the given equation: \(\frac{1}{27} = x^{-3}\).
2. Recall that a negative exponent means the reciprocal, so rewrite \(x^{-3}\) as \(\frac{1}{x^3}\), giving \(\frac{1}{27} = \frac{1}{x^3}\).
3. Since the fractions are equal and both have numerator 1, set the denominators equal: \$27 = x^3$.
4. To solve for \(x\), take the cube root of both sides: \(x = \sqrt[3]{27}\).
5. Simplify the cube root to find the value of \(x\).

Key Concepts

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, x^(-n) equals 1 divided by x^n. Understanding this allows rewriting expressions like x^-3 as 1/x^3.

Properties of Exponents

Exponent rules, such as a^(m) * a^(n) = a^(m+n) and (a^m)^n = a^(m*n), help simplify and solve equations involving powers. Applying these properties enables manipulation of expressions to isolate variables.

Solving Exponential Equations

Solving equations with variables in exponents often involves rewriting both sides with the same base or using logarithms. In this problem, expressing both sides as powers of 3 allows equating exponents to find x.