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

Verified step by step guidance

1. First, recognize that the expression $({0.1}^{3/2})({90}^{3/2})$ can be simplified by using the property of exponents $(a^m)(b^m)=(ab{)}^m$.
2. Combine the bases under a single exponent: $(0.1\times 90{)}^{3/2}$.
3. Calculate the product inside the parentheses: $0.1\times 90=9$.
4. Now, the expression becomes $9^{3/2}$.
5. To evaluate $9^{3/2}$, recognize that it is equivalent to $(9^{1/2}{)}^3$, which means finding the square root of 9 and then cubing the result.

Key Concepts

Exponents

Exponents represent repeated multiplication of a number by itself. For example, in the expression 0.1^3, the base 0.1 is multiplied by itself three times, resulting in 0.001. Understanding how to manipulate exponents is crucial for simplifying expressions, especially when dealing with fractional or negative bases.

Order of Operations

The order of operations is a set of rules that dictates the sequence in which mathematical operations should be performed to ensure consistent results. The common acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) helps remember this order. Applying these rules correctly is essential for accurately calculating expressions like (0.1^3/2)(90^3/2).

Mental Math Techniques

Mental math techniques involve strategies for performing calculations in your head without the use of a calculator or paper. Techniques such as breaking numbers into simpler components, using approximations, and recognizing patterns can significantly speed up calculations. For example, simplifying 90^3/2 can be approached by first calculating 90^3 and then dividing by 2 mentally.