Calculate each value mentally. (0.2^2/3)(40^2/3)

Verified step by step guidance

1. Recognize that the expression is a product of two terms with fractional exponents: \((0.2^{2/3})(40^{2/3})\).
2. Use the property of exponents that states \(a^m \times b^m = (a \times b)^m\) to combine the terms: \((0.2 \times 40)^{2/3}\).
3. Multiply the bases inside the parentheses: \(0.2 \times 40 = 8\).
4. Rewrite the expression as \$8^{2/3}$, which means the cube root of 8 squared.
5. Calculate the cube root of 8, which is 2, and then square it to get the final value.

Key Concepts

Fractional Exponents

Fractional exponents represent roots and powers simultaneously. For example, an exponent of 2/3 means you take the cube root of a number and then square the result. Understanding how to interpret and manipulate fractional exponents is essential for simplifying expressions like (0.2^(2/3)).

Properties of Exponents

The properties of exponents allow us to simplify expressions involving powers, such as the product rule: a^m * a^n = a^(m+n). When bases differ but exponents are the same, like (a^r)(b^r), it can be rewritten as (ab)^r. This helps simplify (0.2^(2/3))(40^(2/3)) into a single term.

Mental Math Strategies for Exponents

Mental math with exponents involves recognizing patterns and simplifying expressions without a calculator. Breaking down numbers into factors or rewriting them in terms of powers of simpler numbers helps. For example, expressing 40 as 8*5 and using exponent rules can make mental calculation easier.