Problem 256 - Elementary Algebra

Question

In the following exercises, simplify. (a) \((\sqrt{7})^{2}\) (b) \((-\sqrt{15})^{2}\)

Step-by-Step Solution

Answer

a) 7, b) 15

Step 1: Simplify the first expression

The given expression is \((\sqrt{7})^2\). To simplify \((\sqrt{a})^2\), where \((\sqrt{a})\) represents the square root of \((_a\)), use the rule that \((\sqrt{a})^2 = a\). Therefore, \((\sqrt{7})^2 = 7\).

Step 2: Simplify the second expression

The given expression is \((-\sqrt{15})^2\). First, note that the square of a negative number is positive. Hence, \((-a)^2 = a^2\). Therefore, \((-\sqrt{15})^2 = (\sqrt{15})^2\). Using the rule from the previous step, \((\sqrt{15})^2 = 15\).

Key Concepts

Square Roots

Square roots are special mathematical functions represented by the symbol \(\text{√}\). The square root of a number \(a\) is a value \(b\) such that \(b^2 = a\). In other words, it's a number that, when multiplied by itself, gives the original number. For example, \(\text{√9} = 3\) because \(3 \times 3 = 9\). Similarly, \(\text{√25} = 5\). Here are a few key points about square roots:

• √a is a number x such that \(x \times x = a\).
• The square root of a perfect square is an integer (e.g., \( √16 = 4 \)).
• The square root of a non-perfect square is an irrational number (e.g., \(√7\) or \(√15\)).

To simplify expressions with square roots, you often need to square or multiply the root. For example, in exercise (a), you have \(\(\text{√7}^2\) = 7\). This is because squaring the square root of 7 returns the original number 7.

Exponents

Exponents are used to represent repeated multiplication of the same number by itself. For example, \(2^3 = 2 \times 2 \times 2 = 8\). The general form of an exponent is \(a^n\), where \(a\) is the base and \(n\) is the exponent. Exponents follow specific rules, including:

• Multiplication rule: \(a^m \times a^n = a^{m+n}\)
• Division rule: \(a^m / a^n = a^{m-n}\)
• Power of a power: \((a^m)^n = a^{m \times n}\)

This last rule is relevant for the given exercises. In Exercise (b), we see \(\((-\text{√15})^2\) \), which applies the power rule to simplify to the base term. Importantly, squaring a negative value results in a positive number, as \(\((-a)^2 = a^2\)\). Thus, \(\((-\text{√15})^2\) = (\text{√15})^2 = 15\).

Simplification Rules

Simplifying expressions involves applying various mathematical rules to reduce them to simpler forms. Here are some common simplification rules:

• Combining like terms: Terms with the same variable and exponent can be added or subtracted.
• Using exponent rules: Apply rules such as power of a power, product of powers, and quotient of powers to simplify expressions involving exponents.
• Simplifying roots: Combine or separate square roots where possible, and remember that \(\(\text{√}a \times \text{√}b = \text{√}(a \times b)\)\).

Specifically, for square roots and exponents, remember:

• Squaring a square root: \(\((\text{√}a)^2 = a\)\).
• Handling negative roots: When squaring a negative square root, the negative sign doesn't affect the result because \(\((-a)^2 = a^2\)\).

Applying these principles lets us simplify expressions like those in the provided exercises effectively.