Problem 259 - Elementary Algebra

Question

In the following exercises, simplify. (a) \((\sqrt{23})^{2}\) (b) \((-\sqrt{3})^{2}\)

Step-by-Step Solution

Answer

(a) 23, (b) 3

Step 1: Simplify \(\sqrt{23}^{2}\)

To simplify \(\sqrt{23}^{2}\), use the property that \(\sqrt{a}^{2} = a\). Therefore, \(\sqrt{23}^{2} = 23\).

Step 2: Simplify \((-\sqrt{3})^{2}\)

To simplify \((-\sqrt{3})^{2}\), use the property that \(a^2 = a \cdot a\). Thus, \((-\sqrt{3})^{2} = (-\sqrt{3}) \cdot (-\sqrt{3})\). Since a negative times a negative is positive, this simplifies to \(\sqrt{3} \cdot \sqrt{3} = 3\). Therefore, \((-\sqrt{3})^{2} = 3\).

Key Concepts

radical expressions

Radical expressions involve roots, such as square roots, cube roots, and so on. The most common radical is the square root, denoted by the symbol \(\backslashsqrt{...}\). For example, the square root of 16 is written \(\backslashsqrt{16}\), and it equals 4 because \({4}\times{4}=16\). Simplifying radical expressions often involves reducing them to their simplest form. This might mean finding an equivalent expression without radicals or simplifying the radical itself. For instance, \(\backslashsqrt{50}\) can be simplified to \(\backslashsqrt{25 \times 2}\) which is \(\backslashsqrt{25}\ \times \sqrt{2} \), simplifying further to \({5}\backslashsqrt{2}\). Practicing these simplifications helps you handle radicals with ease.

squared radicals

When dealing with squared radicals, understanding the relationship between exponents and roots is key. Squaring a square root effectively cancels out the radical, leading you back to the original number under the root symbol. Mathematically, this is represented as \(\backslashsqrt{a}^{2} = a\). For example, \(\backslashsqrt{23}^{2}\) simplifies directly to 23. This is because squaring undoes the square root. When you have an expression like \((-\backslashsqrt{3})^{2}\), you first square the negative sign and the square root separately. A negative squared becomes positive, so \({(- \backslashsqrt{3})^{2}} = \backslashsqrt{3} \backslashcdot \backslashsqrt{3} = 3\). It's helpful to remember that the square of a square root simplifies beautifully to the number inside the root.

properties of exponents

Understanding the properties of exponents is crucial when simplifying expressions involving roots and powers. Exponents tell you how many times to multiply a number by itself. For example, \({a^2 = a \times a}\). Some useful properties include:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m\times n}\)
• Power of a Product: \((a \cdot b)^n = a^n \cdot b^n\)

When simplifying expressions with exponents, it's often useful to apply these properties. For example, in the expression \((-\backslashsqrt{3})^2\), you can interpret it as \( \backslashleft(-1 \backslashcdot \backslashsqrt{3}\backslashright)^2 = (-1)^2 \backslashcdot \backslashsqrt{3}^2 = 1 \backslashcdot 3 = 3\). Practicing these exponent properties helps in simplifying complex expressions easily.