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Chapter 8: Problem 20

Simplify. $$\sqrt{8} \sqrt{32}$$

Short Answer

Expert verified
16

Step by step solution

01

Write the Expression as a Single Square Root

Combine the square roots under a single radical: ewline \ \ \ \[ \ \ \sqrt{8} \ \sqrt{32} \ = \ \sqrt{8 \ \cdot 32} \ \]
02

Multiply Inside the Square Root

Calculate the product under the radical sign: ewline \ \ \[ \ \sqrt{8 \ \cdot 32} = \ \sqrt{256} \ \]
03

Simplify the Square Root

Take the square root of 256: ewline \ \ \[ \ \sqrt{256} = 16 \ \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Roots
A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because 3 multiplied by 3 equals 9. Square roots are commonly represented by the radical symbol \( \sqrt{} \).

Numbers can have positive and negative square roots because both positive and negative numbers, when squared, result in positive values. However, we usually consider only the principal (positive) square root in most mathematical contexts.
Multiplication under Radical
Multiplying numbers under a single square root is often easier to manage. The product rule for square roots states that the square root of a product of two numbers is equal to the product of their individual square roots. Mathematically, this is represented as: \[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \]

For example, multiplying \( \sqrt{8} \) and \( \sqrt{32} \) can be combined under one radical:

\( \sqrt{8} \cdot \sqrt{32} = \sqrt{8 \cdot 32} \)

After combining, you only need to multiply the numbers under the radical sign, making your calculations simpler.
Simplification of Radicals
Simplifying radicals will give you a cleaner and more understandable result. To simplify a square root, you first find the prime factors of the number inside the radical and then pair the prime factors. Two identical factors can be taken out of the radical as a single number. Let's simplify \( \sqrt{256} \):
  • Prime factorization of 256 is \( 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^8 \).
  • Pair the prime factors: \( (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) = 4 \times 4 \times 4 \times 4 = 16 \).

The square root of 256 simplifies to 16 because \( 16 \times 16 = 256 \). This method ensures your final answer is in its simplest form.

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