In Exercises 81–100, evaluate or simplify each expression without using a calculator. 10^{(log ∛x)}

Verified step by step guidance

1. Recognize that the expression is \(10^{\log \sqrt[3]{x}}\), where \(\log\) denotes the logarithm base 10.
2. Recall the property of logarithms and exponents: \(a^{\log_a b} = b\). Here, the base of the exponent and the base of the logarithm are both 10.
3. Rewrite the expression using the property: \(10^{\log \sqrt[3]{x}} = \sqrt[3]{x}\).
4. Express the cube root as a fractional exponent: \(\sqrt[3]{x} = x^{\frac{1}{3}}\).
5. Conclude that the simplified form of the expression is \(x^{\frac{1}{3}}\).

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the power rule: log(a^b) = b * log(a). Understanding these properties allows you to manipulate and simplify logarithmic expressions effectively.

Exponent and Radical Relationships

Radicals can be expressed as fractional exponents, for example, the cube root of x is x^(1/3). Recognizing this helps convert roots into exponents, making it easier to apply logarithmic and exponential rules.

Inverse Relationship Between Exponents and Logarithms

Exponents and logarithms are inverse operations, meaning that a^(log_a(x)) = x. This fundamental relationship is key to simplifying expressions like 10^(log ∛x), where the base of the exponent and the logarithm match.