Question

$$\sqrt{16}=$$____ : $$\sqrt{(-4)^{2}}= $$_____

Short answer

The square root of 16 is 4. The square root of \((-4)^{2}\) is also 4.

Step-by-step solution

- Simplify the first expression

To find the value of \(\function{sqrt}{16}\), determine the square root of 16. The square root of a number is the value that, when multiplied by itself, gives the original number. Here, \(\function{sqrt}{16}\) = 4 because 4 \(times\) 4 = 16.

- Simplify the second expression inside the square root

First, simplify \((-4)^{2} \). Squaring a number means multiplying the number by itself. So, \((-4)^{2} \) means \((-4) \times (-4) \), which equals 16.

- Take the square root of the simplified expression

Now, find the square root of the result from Step 2. This means taking \(\function{sqrt}{16}\), which, as determined in Step 1, is 4.

Key concepts

square roots

A square root of a number is a value which, when multiplied by itself, gives the original number. For example, \( \sqrt{16} \) asks for a number which times itself equals 16. Here, both 4 and -4 are valid because \( 4 \times 4 = 16 \) and \( -4 \times -4 = 16 \). However, we usually refer to the positive square root when we mention the square root of a number. Thus, \( \sqrt{16} = 4 \). Remember the properties of square roots: the square root of zero is zero, and square roots of negative numbers are not real, because you cannot square a real number to get a negative number.

exponents

An exponent tells you how many times to multiply the base number by itself. In \( (-4)^{2} \), the base is -4, and the exponent is 2. To evaluate this, you multiply -4 by itself: \( (-4) \times (-4) = 16 \). Notice that because we are squaring the negative number within parentheses, the negative sign also gets squared, turning it positive. Without parentheses, the calculation would follow different rules where only the number is squared and the negative sign remains outside. For both even exponents (e.g., 2, 4, 6), squaring a negative number results in a positive number.

simplification

Simplification is about breaking down expressions to their most straightforward form. Consider \(\sqrt{(-4)^{2}} \). Follow these steps:
1. Simplify the inner expression \( (-4)^{2} = 16 \).
2. Take the square root of the result: \( \sqrt{16} = 4 \). Simplifying expressions helps in solving problems more easily by focusing on manageable parts. With practice, identifying steps gets easier, making complex problems less daunting.