Problem 18 - Algebra Form and Function

Question

Can the expression be written in the form \(k x^{p}\) ? If so, give the values of \(k\) and \(p\). $$ \frac{1}{3 x^{2}} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(\frac{1}{3x^2}\) in the form \(kx^p\) and identify the values of \(k\) and \(p\). Answer: The expression can be rewritten as \(\frac{1}{3}x^{-2}\), where \(k = \frac{1}{3}\) and \(p = -2\).

Step 1: Rewrite the expression in the form \(kx^p\)

Given the expression \(\frac{1}{3x^2}\), we can rewrite it in the desired form by bringing the term \(x^2\) to the numerator. To do this, we recall that the reciprocal of a power is equivalent to changing the sign of the exponent. Therefore, we will have: $$ \frac{1}{3x^2} = \frac{1}{3} \cdot \frac{1}{x^2} = \frac{1}{3} \cdot x^{-2} $$ Now, we have our expression in the form \(kx^p\), where \(k\) is a constant and \(p\) is an exponent.

Step 2: Identify values of \(k\) and \(p\)

We can now compare the rewritten expression with the general \(kx^p\) form: $$ \frac{1}{3} x^{-2} = kx^p $$ From the comparison, it is clear that the two expressions are equal if \(k = \frac{1}{3}\) and \(p = -2\). Thus, the given expression can indeed be written in the form \(kx^p\), with \(k = \frac{1}{3}\) and \(p = -2\).

Key Concepts

Exponents

Exponents are a fundamental concept in algebra and are used to simplify expressions. They are essentially shorthand for expressing a number multiplied by itself a certain number of times. For instance, in the expression \(x^2\), the exponent \(2\) tells us to multiply \(x\) by itself, resulting in \(x \cdot x\).
This idea extends to negative exponents, where the negative sign indicates a reciprocal. That means instead of multiplying \(x\), you're dividing by it.
For example, \(x^{-2}\) is the same as \(1/x^2\), demonstrating that negative exponents help simplify expressions by showing how they can be inverted.

Polynomial Form

In algebra, the polynomial form is a way of expressing expressions as sums of constants multiplied by powers of a variable. This standard form helps in simplifying, comparing, and performing arithmetic on algebraic expressions.
Polynomials take the form \(a_nx^n + a_{n-1} x^{n-1} + ... + a_1x + a_0\), where each \(a\) is a coefficient, and \(n\) denotes the power of the variable \(x\).
Even expressions that involve division, such as \(\frac{1}{3x^2}\), can sometimes be rewritten in polynomial form through exponent rules. In our example, it becomes \(\frac{1}{3}x^{-2}\), fitting the polynomial template of \(kx^p\) with \(k=\frac{1}{3}\) and \(p=-2\). Utilizing this form provides a unified approach to analyze the structure and behavior of algebraic expressions.

Reciprocal

The reciprocal is simply the flipped version of a fraction or number. For any number \(a\), its reciprocal is \(1/a\). With variables, this concept becomes crucial in algebraic manipulations such as simplifying fractions containing variables.
When dealing with powers, the reciprocal changes the nature of the exponent. A positive exponent becomes negative, signaling that the number or variable is now in the denominator.
Take, for example, \(x^2\). Its reciprocal is \(1/x^2\) or, equivalently, \(x^{-2}\). Understanding reciprocals is key to transforming and simplifying expressions, allowing terms on the denominator to be efficiently expressed in the numerator using negative exponents.