Calculate each value mentally. (0.25³)(400³)

Verified step by step guidance

1. Recognize that the expression is a product of two cubes: \((0.25)^3\) and \((400)^3\).
2. Use the property of exponents that states \((a)^3 \times (b)^3 = (a \times b)^3\) to combine the terms into a single cube: \((0.25 \times 400)^3\).
3. Calculate the product inside the parentheses: multiply \$0.25\( by \)400$.
4. Rewrite the expression as the cube of the result from step 3: \((\text{result})^3\).
5. Finally, cube the number obtained in step 4 by multiplying it by itself three times.

Key Concepts

Exponentiation and Powers

Exponentiation involves raising a base number to a power, which means multiplying the base by itself repeatedly. For example, 0.25³ means 0.25 × 0.25 × 0.25. Understanding how to compute powers is essential for simplifying expressions involving exponents.

Properties of Exponents

The properties of exponents, such as (a^m)(b^m) = (ab)^m, allow us to simplify expressions by combining bases raised to the same power. This property helps in rewriting (0.25³)(400³) as (0.25 × 400)³, making mental calculation easier.

Mental Math Strategies

Mental math strategies involve simplifying numbers and using known facts to calculate values quickly without paper. Recognizing that 0.25 × 400 equals 100 allows you to compute (100)³ mentally, which is simpler than calculating each power separately.