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, so this property applies.
3. Rewrite the expression inside the logarithm: \(\sqrt[3]{x} = x^{\frac{1}{3}}\).
4. Apply the logarithm power rule: \(\log(x^{\frac{1}{3}}) = \frac{1}{3} \log x\).
5. Use the exponent and logarithm property to simplify: \(10^{\log \sqrt[3]{x}} = 10^{\frac{1}{3} \log x} = (10^{\log x})^{\frac{1}{3}} = x^{\frac{1}{3}}\).

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 rewrite and simplify complex logarithmic expressions effectively.

Relationship Between Exponents and Logarithms

Exponents and logarithms are inverse operations. For example, 10^(log x) = x when the log base is 10. This inverse relationship helps simplify expressions where an exponent is a logarithm.

Simplifying Radicals and Fractional Exponents

Radicals like ∛x can be expressed as fractional exponents (x^(1/3)). Converting radicals to fractional exponents makes it easier to apply logarithmic and exponential rules during simplification.