Problem 27 - Algebra Form and Function

Question

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

Step-by-Step Solution

Answer

Answer: k = \(\frac{1}{8}\) and p = \(-\frac{3}{2}\).

Step 1: Simplify the expression

First, we will simplify the given expression by raising the fraction and the term inside the parentheses to the power of \(3\). $$ \left(\frac{1}{2 \sqrt{x}}\right)^{3} = \frac{1^3}{(2\sqrt{x})^3} $$

Step 2: Calculate the fraction's denominator

Now, let's focus on the denominator. The cube of \(2\sqrt{x}\) can be written as: $$ (2\sqrt{x})^3 = 2^3(\sqrt{x})^3 = 8x^{3/2} $$

Step 3: Rewrite the expression

We can now rewrite the expression as: $$ \frac{1^3}{(2\sqrt{x})^3} = \frac{1}{8x^{3/2}} $$

Step 4: Determine the values of \(k\) and \(p\)

Since the expression can be written in the form \(kx^p\), where the values we're looking for are \(k = \frac{1}{8}\) and \(p = -\frac{3}{2}\). So, the expression can be written in the form \(k x^{p}\) as \(\frac{1}{8}x^{-3/2}\).

Key Concepts

Exponent Laws

Exponent laws are fundamental rules that help us simplify expressions involving powers. These laws make operations with exponents much more straightforward:

• Power of a Power: \( (a^m)^n = a^{m \times n} \). To raise a power to another power, multiply the exponents.
• Power of a Product: \( (ab)^n = a^n b^n \). To raise a product to a power, raise each factor to the power separately.
• Power of a Fraction: \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \). To raise a fraction to a power, apply the power to both the numerator and the denominator.

In this exercise, we use these laws to simplify \( \left(\frac{1}{2 \sqrt{x}}\right)^3 \) by applying the power of a fraction rule, resulting in the expression \( \frac{1^3}{(2\sqrt{x})^3} \). This step is crucial for breaking down complex expressions into manageable parts.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. Here, our goal is to make expressions easier to work with. With exponents, simplification often involves reducing powers and applying arithmetic to coefficients.

For example, after using the exponent laws, the denominator \( (2\sqrt{x})^3 \) of our original expression becomes \( 2^3(\sqrt{x})^3 = 8x^{3/2} \). By seeing \( \sqrt{x} \) as \( x^{1/2} \), this makes it easier to apply the power of a power rule and simplifies to \( x^{3/2} \).

The expression is now \( \frac{1}{8x^{3/2}} \), demonstrating that by simplifying, we turn a daunting expression into a form where calculations and comparisons become more straightforward.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations that signify a particular calculation or set of calculations. These expressions can often be rewritten in different forms to reveal new insights or to make solving problems easier.

In this exercise, rewriting the given expression into the form \( kx^p \) involves identifying constants and variables that conform to a recognized pattern. The simplified expression \( \frac{1}{8x^{3/2}} \) can be rearranged to match \( kx^p \), where \( k = \frac{1}{8} \) and \( p = -\frac{3}{2} \), revealing the interaction between the coefficient and the variable's power.

Understanding mathematical expressions in this way helps us manipulate them to solve equations, compare quantities, or express phenomena in different mathematical areas. This knowledge is essential in various fields within mathematics and beyond.