Problem 258 - Elementary Algebra

Question

In the following exercises, simplify. a) \((\sqrt{19})^{2}\) ( b) \((-\sqrt{5})^{2}\)

Step-by-Step Solution

Answer

(a) 19(b) 5.

Step 1 Title - Simplify \( (\sqrt{19})^{2} \)

Recall that when you square a square root, the two operations cancel each other out. Mathematically, \( (\sqrt{a})^{2} = a \). Therefore, \((\sqrt{19})^{2}\) simplifies to 19.

Step 2 Title - Simplify \( (-\sqrt{5})^{2} \)

Here, also remember that squaring a number, whether it's positive or negative, will result in a positive number. Thus, \( (-\sqrt{5})^{2} = (-\sqrt{5}) \times (-\sqrt{5}) \). This simplifies to \[ (-\sqrt{5})^{2} = (\sqrt{5})^{2} = 5 \].

Key Concepts

Properties of Exponents

Exponents are incredibly powerful tools in mathematics. They help us simplify expressions and solve equations more easily. Some key properties can help you work with them.

• Product of Powers: When you multiply two powers with the same base, you add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Power of a Power: To raise a power to another power, you multiply the exponents. For instance, \( (a^m)^n = a^{m \times n} \).
• Power of a Product: To raise a product to a power, raise each factor to that power. That is, \( (ab)^n = a^n b^n \).
• Zero Exponent: Any non-zero number raised to the zero power equals one, \( a^0 = 1 \) for \(a eq 0\).
• Negative Exponent: A negative exponent represents a reciprocal. For example, \( a^{-n} = \frac{1}{a^n} \).

Understanding these properties makes it easier to simplify complex expressions and solve equations.

Squaring Square Roots

The concept of squaring square roots often confuses students, but it's simpler than you may think.

• Square root and squaring are inverse operations. This means that when you square a square root, they cancel each other out.
• Mathematically, \( (\text{sqrt}(a))^2 = a \). For instance, \( (\text{sqrt}(19))^2 = 19 \).
• This also applies to negative square roots. Always remember that squaring a negative square root will still give a positive result since \( (-\text{sqrt}(b))^2 = (\text{sqrt}(b))^2 = b \).

Applying this to our exercise, you get:
\( (\text{sqrt}(19))^2 = 19 \)
and
\( (-\text{sqrt}(5))^2 = (\text{sqrt}(5))^2 = 5 \). This concept is straightforward because squaring undoes the square root.

Negative Numbers

Understanding how negative numbers interact with mathematical operations is crucial.

• When you multiply or square a negative number, you get a positive result. For example, \( (-a) \times (-a) = a^2 \). This happens because multiplying two negatives results in a positive.
• But keep in mind that if you add or subtract negative numbers, the result depends on their signs. For instance, \( -a - b = -(a+b) \) and \( -a + b = b - a \).
• In this exercise, we saw: \( (-\text{sqrt}(5)) \times (-\text{sqrt}(5)) = (\text{sqrt}(5))^2 = 5 \).

Working through these examples helps solidify understanding of how negative numbers behave. This knowledge is fundamental in algebra and higher-level math.