Problem 57 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Give the exact value if possible. Otherwise, approximate to the nearest hundredth. $$ -\sqrt{49} $$

Step-by-Step Solution

Answer

-7

Step 1: Evaluate the square root

First, need to evaluate the square root of 49. The square root of 49 is 7 since 7*7 equals to 49.

Step 2: Consider the negative sign

The original expression specifies that the square root of 49 is to be negated. Therefore, add a negative sign to the result of the square root. So, -7 is the result.

Key Concepts

Understanding Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. When we see the square root symbol, which looks like \( \sqrt{} \), we're looking for a number that fits this criteria. In our exercise, \( \sqrt{49} \) asks us to find a number that, when squared, equals 49.

As explained in the step-by-step solution, \( 7 \times 7 = 49 \), so \( 7 \) is the square root of 49. It's also worth noting that every positive number has two square roots: a positive and a negative one, because \( (-7) \times (-7) = 49 \) as well. But when you see the square root symbol without any additional notation, it typically refers to the 'principal square root,' which is the positive one. In this case, the exercise specifically requires the negative root, hence the solution is \( -7 \).

Working with Negative Numbers

Negative numbers are the numbers that are less than zero. They represent a lack or a loss of something and are denoted by a minus sign \( - \). In the context of square roots, a negative square root is simply the negative value of the positive square root. For example, while the positive square root of 49 is \( +7 \), the negative square root is \( -7 \).

Remember, squaring a negative number, like any other number, gives a positive result because you're multiplying the number by itself. Thus, \( -7 \times -7 \) equals \( +49 \), a key concept when dealing with the square roots of positive numbers.

Approximating Numbers

Approximating numbers involves finding a value that is close to the exact value, often to a specific degree of accuracy, such as to the nearest hundredth. We usually approximate when the exact value is difficult or impractical to determine, or when it's an irrational number that cannot be expressed as a precise fraction or decimal.

In the given exercise, approximation is only necessary if the square root did not result in a whole number, which often happens with non-square numbers. To approximate, one would typically use methods like finding the nearest squares or using a calculator. Since \( \sqrt{49} \) is an exact whole number, there's no need for approximation in this case. However, if the exercise had involved a number like 45, you would approximate \( \sqrt{45} \) to the nearest hundredth (for example, \( \approx 6.71 \) when rounded to two decimal places).