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

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Rewrite the expression using fractional exponents: $(0.1^{\frac{3}{2}})(90^{\frac{3}{2}})$.

Recall the property of exponents that states $a^m \cdot b^m = (a \cdot b)^m$. Apply this to combine the terms: $(0.1 \cdot 90)^{\frac{3}{2}}$.

Multiply the bases inside the parentheses: $0.1 \times 90 = 9$; so the expression becomes $9^{\frac{3}{2}}$.

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 $\left(\sqrt{9}\right)^3$.

Calculate the square root of 9, then raise the result to the third power to find the value of the expression.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Textbook Question

Factor by any method. See Examples 1–7. $4 z^{4} - 7 z^{2} - 15$