Problem 80 - Algebra 2

Question

a. Reasoning Show that \(\forall f / x^{2}=\sqrt{x}\) by using the definition of fourth root. b. Reasoning Show that \(\sqrt[4]{x^{2}}=\sqrt{x}\) by rewriting \(\sqrt[4]{x^{2}}\) in exponential form.

Step-by-Step Solution

Answer

The assertion in part (a) of the exercise, \(x^{2} = \sqrt{x}\), is incorrect. However, in part (b), \( \sqrt[4] {x^{2}} = \sqrt{x}\), as shown by rewriting the fourth root of \(x^{2}\) into exponential form.

Step 1: Part (a) Step 1: Recognize the definitions

Recall that the fourth root of a number \(x\) is \(x^{1/4}\). So, \(\sqrt{x} = x^{1/2}\). Therefore, to prove that \(x^{2} = \sqrt{x}\), we must show that \(x^{2} = x^{1/2}\).

Step 2: Part (a) Step 2: Contradiction

Observing the two expressions \(x^{2}\) and \(x^{1/2}\), it becomes clear that they are not equivalent, because squaring a number (\(x^{2}\)) is not the same operation as taking the square root of a number (\(\sqrt{x} = x^{1/2}\)). So, the assertion that \(x^{2} = \sqrt{x}\) for all \(x\) holds incorrect.

Step 3: Part (b) Step 1: Rewrite in exponential form

Recalling the relationship between root and exponent form: \(\sqrt[n]{a^m} = a^{m/n}\) where \(a\) is a positive real number, \(n\) and \(m\) are integers with \(n>0\), rewrite \(\sqrt[4]{x^{2}}\) in exponent form to get \(x^{2/4}\).

Step 4: Part (b) Step 2: Simplify the exponent

Simplify the fraction in the exponent 2/4 to get 1/2. Substituting back into the equation gives \(x^{1/2}\), which is the definition of \(\sqrt{x}\). Hence, \(\sqrt[4]{x^{2}} = \sqrt{x}\).

Key Concepts

Fourth Root

Understanding the fourth root can help simplify expressions in algebra. When we talk about the fourth root of a number, we refer to the value that, when raised to the fourth power, gives us the original number.

To express this in mathematical terms, the fourth root of a number, say \( x \), is written as \( \sqrt[4]{x} \) or equivalently as \( x^{1/4} \).

• This concept is similar to the square root, which is the second root and represented by \( x^{1/2} \).
• For example, if \( x = 16 \), then \( \sqrt[4]{16} = 2 \) because \( 2^4 = 16 \).

Recognizing how different powers of a number relate helps in understanding more complex expressions and equations. This notion of roots can be extended to any positive integer, not just four.

Exponential Form

Exponential form is a way to write numbers using exponents, making it easier to manage large or complex calculations in algebra.

For roots, we can transform the notation of a root into exponential form using the equation \( \sqrt[n]{a^m} = a^{m/n} \). For example, if we have a fourth root \( \sqrt[4]{x^2} \), we can rewrite it in exponential form as \( x^{2/4} \).

• This transformation is beneficial for simplifying expressions.
• Understanding exponent rules is key; for example, \( \sqrt{x} \) can also be seen as \( x^{1/2} \).

This form allows algebraic manipulation by making the operations more straightforward, especially when simplifying expressions or solving equations.

Algebraic Simplification

Algebraic simplification involves rewriting expressions in a simpler or more understandable form while preserving their value. This is crucial in algebra as it makes handling complex expressions easier.

To simplify an expression like \( \sqrt[4]{x^2} \), we reformulate it in exponential terms, \( x^{2/4} \), which can then be simplified further to \( x^{1/2} \) or \( \sqrt{x} \).

• The simplification process involves reducing fractions in exponents, like turning \( 2/4 \) to \( 1/2 \).
• By doing this, the operation becomes easier to understand and compute.

Simplifying expressions, especially those involving roots and exponents, helps solve equations more efficiently and reduces the likelihood of making errors.