Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. 100^3/2

Verified step by step guidance

1. Recognize that the expression \$100^{3/2}$ is an exponent with a fractional power, where the numerator (3) is the power and the denominator (2) is the root.
2. Rewrite the expression using the property of fractional exponents: \(a^{m/n} = \left( a^{m} \right)^{1/n} = \sqrt[n]{a^{m}}\). So, \$100^{3/2} = \left( 100^{3} \right)^{1/2}\( or equivalently \)\sqrt{100^{3}}$.
3. Calculate the inner exponent first: \$100^{3} = 100 \times 100 \times 100\( (you can leave it as \)100^{3}$ for now if you prefer to simplify under the root).
4. Take the square root of the result: \(\sqrt{100^{3}} = \sqrt{(100^{2} \times 100)} = \sqrt{100^{2}} \times \sqrt{100}\) using the property \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\).
5. Simplify the square roots: \(\sqrt{100^{2}} = 100\) and \(\sqrt{100} = 10\), so the expression simplifies to \$100 \times 10$. This gives the simplified form without negative exponents.

Key Concepts

Exponents and Powers

Exponents indicate how many times a base number is multiplied by itself. For example, 100^3 means 100 multiplied by itself three times. Understanding how to manipulate exponents is essential for simplifying expressions involving powers.

Rational Exponents

Rational exponents like 3/2 represent roots and powers combined. Specifically, a^(m/n) means the nth root of a raised to the mth power, e.g., 100^(3/2) equals (√100)^3. This concept helps convert fractional exponents into radical expressions.

Rules for Simplifying Expressions Without Negative Exponents

Negative exponents indicate reciprocals, but the problem requires answers without them. Simplifying involves rewriting expressions to eliminate negative exponents by using reciprocal properties and expressing results with positive exponents only.