Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. 100^3/2

Verified step by step guidance

1. Recognize that the expression is a power of a number: \$100^{3/2}$. This means you are raising 100 to the power of \(\frac{3}{2}\).
2. Recall the property of exponents that \(a^{m/n} = \left(a^{1/n}\right)^m = \left(\sqrt[n]{a}\right)^m\). Here, \(n=2\) and \(m=3\), so rewrite the expression as \(\left(\sqrt{100}\right)^3\).
3. Calculate the square root of 100, which is \(\sqrt{100} = 10\).
4. Now raise 10 to the power of 3, which is \$10^3$.
5. Express the final answer without negative exponents by writing \$10^3\( as \(10 \times 10 \times 10\) or simply keep it as \)10^3$.

Key Concepts

Exponents and Rational Exponents

Exponents indicate how many times a base is multiplied by itself. Rational exponents, like 3/2, represent roots and powers simultaneously; for example, a^(3/2) means the square root of a cubed. Understanding how to interpret and manipulate these is essential for simplifying expressions.

Simplifying Expressions with Exponents

Simplifying expressions involves applying exponent rules such as multiplying powers, raising powers to powers, and converting between radical and exponential forms. This helps rewrite expressions in simpler or more standard forms, especially when removing negative exponents.

Writing Expressions Without Negative Exponents

Negative exponents indicate reciprocals, so expressions with negative exponents can be rewritten as fractions with positive exponents. Since the problem requires answers without negative exponents, converting all negative powers to positive ones is necessary for the final simplified form.