1. **State the problem:** We need to determine which statements about radicals and rational exponents are true. 2. **Recall definitions and properties:** - The nth root of a number $a$ is written as $\sqrt[n]{a}$. - Rational exponents are defined as $a^{p/q} = \left(\sqrt[q]{a}\right)^p$. - The square root of $a$ is $\sqrt{a}$, which is the same as $a^{1/2}$. 3. **Evaluate each statement:** - Statement 1: "The nth root of a can be written as $\sqrt[n]{a}$ and as $a^{1/n}$" - This is true by definition of rational exponents. - Statement 2: "$a^{p/q} = \sqrt[q]{a^q} = \left(\sqrt[q]{a}\right)^q$" - Check carefully: - $a^{p/q} = \left(\sqrt[q]{a}\right)^p$, not $a^{p/q} = \sqrt[q]{a^q}$. - Also, $\sqrt[q]{a^q} = a$, so the middle expression is incorrect. - The last part $\left(\sqrt[q]{a}\right)^q = a$. - So the statement is false. - Statement 3: "$a^{1/n} = \sqrt{a^n}$" - $a^{1/n} = \sqrt[n]{a}$, but $\sqrt{a^n} = (a^n)^{1/2} = a^{n/2}$. - So this is false. - Statement 4: "The notation $\sqrt{a}$ is Radical Notation for the square root of $a$" - This is true. - Statement 5: "The notation $a^{1/2}$ is Rational Exponent Notation for the square root of $a$" - This is true. 4. **Final answer:** - True statements: 1, 4, 5 - False statements: 2, 3