Problem 56 - Elementary Algebra

Question

For the following problems, evaluate each expression. If the expression does not represent a real number, write "not a real number." $$ -(-\sqrt{0.81}) $$

Step-by-Step Solution

Answer

Answer: \(\frac{9}{10}\)

Step 1: Find the square root of 0.81

In order to simplify the expression, we first need to find the square root of 0.81. Remember, the square root is the opposite of squaring a number. In this case, we are looking for a number that, when squared, equals 0.81. It would be helpful to rewrite 0.81 as a fraction: $$ 0.81 = \frac{81}{100} $$ Now we can find the square root: $$ \sqrt{0.81} = \sqrt{\frac{81}{100}} = \frac{\sqrt{81}}{\sqrt{100}} = \frac{9}{10} $$ So the square root of 0.81 is \(\frac{9}{10}\).

Step 2: Negate the negative sign

Now that we have found the square root of 0.81, we can apply the two negative signs in the expression. The two negative signs cancel out each other, since a negative multiplied by a negative equals a positive. Therefore, we can simplify the expression to: $$ -(-\sqrt{0.81}) = \frac{9}{10} $$ So the final result is \(\frac{9}{10}\).

Key Concepts

Square Roots

The concept of square roots is fundamental in algebraic expressions. A square root of a number is essentially the value that, when multiplied by itself, yields the original number. For instance, the square root of 9 is 3 because 3 times 3 equals 9.
When dealing with decimals like 0.81, it's often useful to convert them into fractions to simplify the finding of square roots. In this case, 0.81 is rewritten as \(\frac{81}{100}\).
This fraction then allows us to take the square roots of the numerator and the denominator separately. Thus, \(\sqrt{\frac{81}{100}} = \frac{\sqrt{81}}{\sqrt{100}} = \frac{9}{10}\).

• Knowing how to find the square root of a fraction is an essential skill in algebra.
• The square root symbol (\(\sqrt{}\)) denotes this operation.

Being comfortable with this concept will make evaluating algebraic expressions much easier.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers where the numerator is an integer and the denominator is a non-zero integer. Since these numbers can be written in fraction form, they are very common in mathematical computations, especially in simplifications.
The expression \(\sqrt{0.81}\), when simplified to \(\frac{9}{10}\), is an example of a rational number. The fraction \(\frac{9}{10}\) clearly shows that rational numbers can also include square roots that yield finite decimal or fraction results.

• They can be expressed as fractions. For example, \(0.5\) is \(\frac{1}{2}\).
• Rational numbers include both positive and negative fractions. For example, \(-\frac{3}{4}\) is also a rational number.

Understanding how rational numbers are expressed and simplified from decimals or square roots is key in algebra.

Negation

Negation in algebra involves changing the sign of a number or expression. It's a key part of simplifying expressions, especially when dealing with multiple negative signs. In the original expression \(-(-\sqrt{0.81})\), we have two negatives.
The rule with negatives is that two negatives make a positive. Here, the negation of a negative number turns our expression into a positive \(\frac{9}{10}\). This follows the basic rule:

• A negative sign before a number means the opposite of that number.
• Two negative signs result in a positive, because \(-(-x) = x\).

Being comfortable with this rule helps when simplifying complex expressions.