Calculate each value mentally. (0.1^3/2)(90^3/2)

Verified step by step guidance

1. Rewrite the expression using fractional exponents: \((0.1^{\frac{3}{2}})(90^{\frac{3}{2}})\).
2. Recall the property of exponents that states \(a^m \cdot b^m = (a \cdot b)^m\). Apply this to combine the terms: \((0.1 \cdot 90)^{\frac{3}{2}}\).
3. Multiply the bases inside the parentheses: \(0.1 \times 90 = 9\); so the expression becomes \(9^{\frac{3}{2}}\).
4. Rewrite the fractional exponent \(\frac{3}{2}\) as a combination of a square root and a cube: \(9^{\frac{3}{2}} = (9^{\frac{1}{2}})^3\) or equivalently \(\left(\sqrt{9}\right)^3\).
5. Calculate the square root of 9, then raise the result to the third power to find the value of the expression.

Key Concepts

Exponents and Rational Powers

Rational exponents represent roots and powers combined; for example, a^(m/n) means the nth root of a raised to the mth power. Understanding how to interpret and manipulate these exponents is essential for simplifying expressions like (0.1)^(3/2).

Properties of Exponents

The properties of exponents, such as the product rule a^m * a^n = a^(m+n), allow simplification of expressions involving powers. Recognizing when to apply these rules helps in combining terms like (0.1)^(3/2) and (90)^(3/2) efficiently.

Mental Math Strategies for Exponents

Mental math with exponents involves breaking down numbers into simpler components, such as expressing 0.1 as 1/10 and 90 as 9*10, then applying exponent rules to simplify calculations without a calculator. This approach aids in quick and accurate evaluation.