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

Verified step by step guidance

1. Recognize that the expression is a product of two terms raised to fractional exponents: \((0.1^{3/2})(90^{3/2})\).
2. Recall the property of exponents that allows you to combine terms with the same exponent: \(a^{m} \times b^{m} = (ab)^{m}\). Apply this to rewrite the expression as \((0.1 \times 90)^{3/2}\).
3. Calculate the product inside the parentheses: \$0.1 \times 90$.
4. Rewrite the expression as a single term raised to the power \(\frac{3}{2}\): \((9)^{3/2}\).
5. Understand that the fractional exponent \(\frac{3}{2}\) means you take the square root (denominator 2) and then cube the result (numerator 3), or vice versa. So, calculate \(\sqrt{9}\) first, then raise that result to the power of 3.

Key Concepts

Exponents and Powers

Exponents indicate how many times a base number is multiplied by itself. For example, a^b means multiplying 'a' by itself 'b' times. Understanding how to work with fractional exponents, such as 3/2, is essential for simplifying expressions.

Fractional Exponents

A fractional exponent like 3/2 represents both a power and a root: a^(3/2) equals the square root of a cubed, or (√a)^3. This concept helps in rewriting and simplifying expressions involving roots and powers.

Properties of Exponents in Multiplication

When multiplying expressions with the same exponent, such as (a^m)(b^m), the result can be written as (ab)^m. This property allows combining terms efficiently, which is useful for mental calculation and simplification.