Problem 71 - Intermediate Algebra

Question

Use the quotient rule to divide. Then simplify if possible. See Example 5 $$ \frac{3 \sqrt{100 x^{2}}}{2 \sqrt{2 x^{-1}}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{15x^{3/2}}{\sqrt{2}}\).

Step 1: Simplify the Numerator

The numerator is given by \(3 \sqrt{100 x^2}\). Begin by simplifying inside the square root. Since \(\sqrt{100} = 10\) and \(\sqrt{x^2} = x\), we have: \(3 \times 10 \times x = 30x\). Thus, the simplified numerator is \(30x\).

Step 2: Simplify the Denominator

The denominator is \(2 \sqrt{2 x^{-1}}\). Simplifying inside the square root, we get \(\sqrt{2} \times \sqrt{x^{-1}}\). Since \(\sqrt{x^{-1}} = x^{-1/2}\), the expression becomes \(2 \times \sqrt{2} \times x^{-1/2}\). Thus, the simplified denominator is \(2 \sqrt{2} x^{-1/2}\).

Step 3: Apply the Quotient Rule

The expression to simplify is \(\frac{30x}{2 \sqrt{2} x^{-1/2}}\). The quotient rule says to divide the coefficients and subtract the exponents of like bases. Coefficient division yields \(\frac{30}{2} = 15\). For the variable x, we have \(x^{1 - (-1/2)} = x^{3/2}\). Thus, the simplified expression is \(\frac{15x^{3/2}}{\sqrt{2}}\).

Step 4: Further Simplify if Possible

There are no more like terms or factors that can be combined or simplified further. The expression \(\frac{15x^{3/2}}{\sqrt{2}}\) is in its simplest form.

Key Concepts

Simplify Expressions

Simplifying expressions involves rewriting them in a more straightforward or more readable form. The goal is to reduce complexity while maintaining the original value. In our example, the initial complex fraction was turned into a simpler form by breaking down elements into more manageable parts.

• Simplified expressions have fewer terms.
• They have no parentheses or complex fractions.
• The coefficients and variables are cleanly organized.

To simplify effectively:

• Break down complex terms into prime factors.
• Combine like terms wherever possible.
• Apply arithmetic rules to reduce expressions.

With this practice, you will make expressions clearer and easier to work with, just as we did to move from a complex radical fraction to a clean, simplified quotient of \(\frac{15x^{3/2}}{\sqrt{2}}\).

Radicals

Radicals involve roots, and the most common is the square root. To simplify radicals, we often need to factor them into simpler components. In our exercise:- The radical \(\sqrt{100 x^2}\) was simplified into two parts: the root of a number and the root of a variable.- \(\sqrt{100} = 10\), since 10 is the number that when multiplied by itself gives 100.Understanding radicals helps in recognizing patterns and relationships between numbers and their roots:

• Identify perfect squares within the radical.
• Simplify each part separately before recombining.
• Re-factor variables under the radical to expose simplest forms.

Manipulating radicals correctly is crucial for simplifying expressions effectively, leading us from complicated roots like \(\sqrt{2x^{-1}}\) to straightforward terms like \(x^{-1/2}\).

Exponents

Exponents are potent mathematical tools that signify repeated multiplication. When simplifying expressions, managing exponents is key, especially when they involve variables and radicals.For example, in this exercise:- The exponent \(x^{2}\) was dealt within the numerator where \(x\) was extracted from under the square root.- The term \(x^{-1}\) becomes \(x^{-1/2}\) under the square root sign, illustrating a property where the negative exponent reflects reciprocal action.Key techniques when tackling exponents include:

• When dividing, subtract exponents of like bases, as in \(x^{1 - (-1/2)}\).
• Convert radicals into fractional exponents for straightforward manipulation.
• Turn negative exponents positive by recognizing how they indicate reciprocal action.

Understanding and applying the properties of exponents allows for transforming challenging expressions into simple equivalents.

Algebra Division

Division in algebra often requires the use of the quotient rule, especially with complex expressions. The quotient rule focuses on dividing coefficients and correctly handling exponents.In dividing algebraic expressions like our example, the solution follows these processes:1. **Coefficients Divide First:** Divide the numerical coefficients from the numerator and denominator. For example, \(\frac{30}{2} = 15\).2. **Handle Variables Separately:** Use the rule for exponents \(\frac{a^m}{a^n} = a^{m-n}\) to handle variables individually. This method was used to simplify the division of \(x^1\) by \(x^{-1/2}\).This methodical approach helps in:

• Ensuring clarity and avoiding errors with switching numerator or denominator parts.
• Making complex expressions more manageable by sequential simplification.
• Leading to a reliable form of expression, like \(\frac{15x^{3/2}}{\sqrt{2}}\), without excess terms.

Mastery of algebraic division aids in efficiently breaking down equations and is foundational for solving more complex algebra problems.