Problem 44 - Elementary Algebra

Question

For the following problems, simplify each expressions. $$ \frac{\sqrt{66 m^{12} n^{15}}}{\sqrt{11 m n^{8}}} $$

Step-by-Step Solution

Answer

Question: Simplify the given complex fraction: $$\frac{\sqrt{66 m^{12} n^{15}}}{\sqrt{11 m n^{8}}}$$. Answer: The simplified expression is: $$\sqrt{6} \cdot m^{\frac{11}{2}} \cdot n^{\frac{7}{2}}$$.

Step 1: Rewrite the expression

Rewrite the given expression, separating the radicals in the numerator and the denominator: $$ \frac{\sqrt{66 m^{12} n^{15}}}{\sqrt{11 m n^{8}}} = \frac{\sqrt{66}\sqrt{m^{12}}\sqrt{n^{15}}}{\sqrt{11}\sqrt{m}\sqrt{n^{8}}}, $$

Step 2: Simplify the coefficients

Simplify the coefficients inside the radicals: $$ \frac{\sqrt{66} \cdot \sqrt{m^{12}} \cdot \sqrt{n^{15}}}{\sqrt{11} \cdot \sqrt{m} \cdot \sqrt{n^8}} = \frac{\sqrt{6}\sqrt{11} \cdot \sqrt{m^{12}} \cdot \sqrt{n^{15}}}{\sqrt{11} \cdot \sqrt{m} \cdot \sqrt{n^8}}. $$

Step 3: Simplify the variables

Simplify the variables inside the radicals using the properties of exponents: $$ \frac{\sqrt{6}\sqrt{11} \cdot m^6 \cdot n^{\frac{15}{2}}}{\sqrt{11} \cdot m^{\frac{1}{2}} \cdot n^4}. $$

Step 4: Simplify the expression

Simplify the expression by dividing the radicals and combining like terms: $$ \frac{\sqrt{6}\cancel{\sqrt{11}} \cdot m^6 \cdot n^{\frac{15}{2}}} {\cancel{\sqrt{11}} \cdot m^{\frac{1}{2}} \cdot n^4} = \frac{\sqrt{6} \cdot m^6 \cdot n^{\frac{15}{2}}}{m^{\frac{1}{2}} \cdot n^4}. $$

Step 5: Apply the quotient rule

Apply the quotient rule for exponents to simplify the variables: $$ \frac{\sqrt{6} \cdot m^6 \cdot n^{\frac{15}{2}}}{m^{\frac{1}{2}} \cdot n^4} = \sqrt{6} \cdot m^{6-\frac{1}{2}} \cdot n^{\frac{15}{2} - 4}. $$

Step 6: Simplify the exponents

Simplify the exponents: $$ \sqrt{6} \cdot m^{6-\frac{1}{2}} \cdot n^{\frac{15}{2} - 4} = \boxed{\sqrt{6} \cdot m^{\frac{11}{2}} \cdot n^{\frac{7}{2}}}. $$

Key Concepts

Properties of Exponents

Exponents are powerful tools in mathematics that allow us to simplify and manipulate expressions efficiently. The properties of exponents help us perform operations on powers and simplify expressions involving them. Here are some of the most important properties:

• Product of Powers Property: When multiplying two expressions with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• Power of a Power Property: When a power is raised to another power, you multiply the exponents: \( (a^m)^n = a^{m\cdot n} \).
• Power of a Product Property: A power applied to a product distributes to each factor: \( (ab)^n = a^n \times b^n \).
• Quotient of Powers Property: When dividing expressions with the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

Understanding these properties will help you simplify expressions involving exponents and are crucial when working with exponential equations. For example, in the step-by-step solution above, the properties of exponents were used specifically to simplify the variables within the radical expressions.

Quotient Rule

The quotient rule is an essential exponent rule that helps in dividing one exponent by another when both have the same base. It is directly derived from the properties of exponents. The rule states:When you divide two powers with the same base, subtract the exponent of the denominator from the exponent of the numerator. Mathematically, this is shown as:\[ \frac{a^m}{a^n} = a^{m-n} \]In the context of simplifying radicals, this means that if you have expressions involving exponents in both the numerator and the denominator, you can simplify by merely subtracting the exponents.In our exercise, the quotient rule was used in Step 5 to simplify the expression further. By subtracting the exponents of like terms in the numerator and denominator, the expression becomes much simpler. This rule is a very powerful tool because it allows for simplification without changing the base, making complex radical expressions more manageable.

Simplifying Expressions

Simplifying mathematical expressions involves reducing them to their simplest form while maintaining their value. When dealing with expressions that contain exponents, radicals, or both, simplifying can make calculations easier and results clearer. There are several strategies to simplify expressions:

• Factor and Cancel: This involves factoring expressions to identify and cancel common terms.
• Use Properties of Exponents: Understanding the laws governing exponents allows you to combine and reduce terms quickly.
• Apply the Quotient Rule: Use this rule to simplify expressions with the same base in both the numerator and the denominator.
• Simplify Coefficients: Any numerical factors outside the variables should be reduced to their simplest form.

In our exercise, these methods were used in various steps to bring the expression to its simplest form. Simplifying is not just about reducing; it's about making expressions easier to work with, especially in more complex equations.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, and can often make a mathematical expression look complex. However, with the right tools, they can be simplified just like any other expression. To work with radical expressions effectively:

• Separate the Radicals: Break down the radical expression into separate parts. In the numerator and denominator, this can make simplification easier.
• Simplify Inside the Radical: First, handle any simplification under the radical sign before simplifying the radical itself.
• Use the Quotient and Product Rules for Radicals: Similar to exponent rules, these can help in breaking down the radical expressions.
• Rationalizing the Denominator: When radicals appear in the denominator, multiplying both the numerator and the denominator by a term that removes the radical from the denominator is often helpful.

In the given problem, radical expressions were made simpler by employing these strategies. Understanding how to manage radicals is essential for simplifying complex mathematical expressions.