Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. 8^2/3

Verified step by step guidance

1. Recognize that the expression is an exponentiation of a number: \$8^{2/3}$. This means you are raising 8 to the power of two-thirds.
2. Recall the rule for fractional exponents: \(a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}\). Here, \(m=2\) and \(n=3\).
3. Apply the cube root first: find \(\sqrt[3]{8}\), which means the number that when cubed gives 8.
4. After finding \(\sqrt[3]{8}\), raise that result to the power of 2, i.e., square it.
5. Write the final expression without any negative exponents, ensuring the answer is simplified completely.

Key Concepts

Exponents and Powers

Exponents indicate how many times a base is multiplied by itself. For example, 8^2 means 8 multiplied by itself twice. Understanding how to manipulate exponents is essential for simplifying expressions involving powers.

Rational Exponents

Rational exponents like 8^(2/3) represent roots and powers combined. The denominator (3) indicates the root (cube root), and the numerator (2) indicates the power. So, 8^(2/3) means the cube root of 8 squared.

Simplifying Expressions Without Negative Exponents

Negative exponents represent reciprocals, but the problem requires answers without them. This means rewriting expressions so all exponents are positive, often by using properties like a^(-n) = 1/a^n.