Problem 68 - Applied Calculus

Question

Write each expression in power form \(a x^{b}\) for numbers \(a\) and \(b\). $$ \frac{10 \sqrt{x}}{2 \sqrt[3]{x}} $$

Step-by-Step Solution

Answer

The expression in power form is \(5x^{1/6}\).

Step 1: Simplify the Division Coefficient

First, simplify the division of the numerical coefficients in the fraction: \(\frac{10}{2} = 5\).

Step 2: Express Radicals as Exponents

Change the radicals in the expression to exponents. The square root expression \(\sqrt{x}\) becomes \(x^{1/2}\).The cube root expression \(\sqrt[3]{x}\) becomes \(x^{1/3}\).

Step 3: Create a Single Power of x

Use the properties of exponents to combine the powers of \(x\). The expression becomes:\[5 \times x^{1/2} \div x^{1/3} = 5 \times x^{1/2 - 1/3}\].

Step 4: Simplify the Exponent

Subtract the exponents: \(\frac{1}{2} - \frac{1}{3}\). Find a common denominator, which is 6:\(\frac{3}{6} - \frac{2}{6} = \frac{1}{6}\).Therefore, the expression becomes \(5x^{1/6}\).

Key Concepts

Radicals as Exponents

When we talk about radicals, such as square roots or cube roots, we're often dealing with numbers or expressions under a radical sign. Rather than keeping them as radicals, these expressions can be rewritten as exponents. Here's how it works:

• For a square root of a variable, like \( \sqrt{x} \), it can be expressed as \( x^{1/2} \). This is because the square root is mathematically equivalent to raising a number to the power of \( \frac{1}{2} \).


• Similarly, the cube root of a variable, such as \( \sqrt[3]{x} \), becomes \( x^{1/3} \). This is done by using the exponent \( \frac{1}{3} \), representing the cube root.

Writing radicals as exponents is particularly handy when working with expressions and equations, as it simplifies combining like terms and applying exponent rules. It turns the problem into one involving familiar arithmetic with numbers and fractions.

Exponent Properties

Exponent properties are essential for simplifying expressions involving powers. Knowing these rules helps you manipulate and combine terms efficiently:

• Product of Powers: When you multiply expressions with the same base, add the exponents. For instance, \( x^a \times x^b = x^{a+b} \).


• Quotient of Powers: When dividing expressions with the same base, subtract the exponents: \( \frac{x^a}{x^b} = x^{a-b} \).


• Power of a Power: When raising an exponent to another power, multiply the exponents: \((x^a)^b = x^{a\cdot b} \).

By applying these properties, we can handle more complex expressions with ease. They allow us to transform an initial expression into something simpler, as with converting and manipulating radicals as exponents.

Simplifying Expressions

Simplifying expressions is all about making them easier to work with. It involves reducing complex forms into simpler ones, often using the exponent properties discussed earlier. Let's break down the steps used in the solution:

• Combine Coefficients: Simplifying begins with addressing any numerical coefficients. For example, in our expression, \( \frac{10}{2} \) simplifies to \( 5 \).


• Adjust Exponents: Once the radicals are expressed as exponents, the term \( \frac{x^{1/2}}{x^{1/3}} \) is simplified by subtracting the exponent in the denominator from that in the numerator, resulting in \( x^{1/2 - 1/3} \).


• Final Expression: By finding a common denominator for the fractions, you get \( x^{1/6} \). Therefore, the simplified form of the original expression is \( 5x^{1/6} \).

By carefully breaking down each part of the expression and applying rules systematically, we achieve a simple, clear result. It becomes much easier to interpret and use in further calculations.