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

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Recognize that the expression is a product of two terms with fractional exponents: \((0.2^{2/3})(40^{2/3})\).

Use the property of exponents that states \(a^m \times b^m = (a \times b)^m\) to combine the terms: \((0.2 \times 40)^{2/3}\).

Multiply the bases inside the parentheses: \(0.2 \times 40 = 8\).

Rewrite the expression as \(8^{2/3}\), which means the cube root of 8 squared.

Calculate the cube root of 8, which is 2, and then square it to get the final value.

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주요 개념

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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.

Calculate each value mentally. (0.22/3)(402/3)