Problem 71 - Algebra and Trigonometry

Question

\(69-82\) . Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. $$ (5 \sqrt[3]{x})(2 \sqrt[4]{x}) $$

Step-by-Step Solution

Answer

The simplified expression is \( 10x^{7/12} \).

Step 1: Express Radicals as Rational Exponents

First, you'll need to convert the radicals into expressions with rational exponents. Recall that the radical expression \( \sqrt[n]{x} \) can be written as \( x^{1/n} \). Therefore, \( \sqrt[3]{x} = x^{1/3} \) and \( \sqrt[4]{x} = x^{1/4} \).

Step 2: Substitute Rational Exponents Into the Expression

Take the expression \( (5 \sqrt[3]{x})(2 \sqrt[4]{x}) \) and replace the radicals with their rational exponent equivalents. This results in the expression: \( (5x^{1/3})(2x^{1/4}) \).

Step 3: Multiply the Coefficients

Multiply the numeric coefficients 5 and 2 together: 5 \( \times \) 2 = 10. Thus, the expression now looks like this: \( 10x^{1/3}x^{1/4} \).

Step 4: Apply the Product of Powers Property

For the expression \( 10x^{1/3}x^{1/4} \), use the product of powers property, which states that when multiplying like bases, you add the exponents: \( x^{1/3} \times x^{1/4} = x^{1/3 + 1/4} \). To add the exponents, find a common denominator: 1/3 = 4/12 and 1/4 = 3/12. Add these fractions to get: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.

Step 5: Write the Simplified Expression

Now, substitute back into the expression with the combined exponent: \( 10x^{7/12} \). This is the simplified form of the given expression with rational exponents.

Key Concepts

Product of Powers Property

The product of powers property is a foundational concept in algebra. It makes multiplying expressions with the same base easier by allowing you to add the exponents together. This property states that for any non-zero number or variable, and for any powers \( a \) and \( b \), the expression \( x^a \times x^b \) simplifies to \( x^{a+b} \). This is because when you multiply like bases, you are essentially combining the repeated multiplication of the base.

• Ensure the bases in your expressions are the same. The product of powers property only applies to expressions where the bases match.
• Make sure the property is applied correctly by keeping track of fractional exponents, as seen in the exercise above.

Using the property correctly in this exercise allowed us to simplify \( x^{1/3} \times x^{1/4} \) into \( x^{7/12} \). When bases are the same, adding the exponents reduces complex expressions into simpler, more manageable ones.

Radical Expressions

Radical expressions often appear intimidating, but they can be easily converted into rational expressions using exponents. A radical expression like \( \sqrt[n]{x} \) can be rewritten as \( x^{1/n} \), where \( n \) is the index of the radical.

• This conversion simplifies the process of performing algebraic operations like multiplication or division involving radicals.
• For the exercise provided, converting \( \sqrt[3]{x} \) to \( x^{1/3} \) and \( \sqrt[4]{x} \) to \( x^{1/4} \) paved the way for further simplification using exponent rules.

It's important to recognize that expressing radicals as rational exponents creates opportunities to apply a variety of algebraic properties like the product of powers property, facilitating the simplification process considerably.

Simplifying Expressions

Simplifying expressions involves reducing them down to their most efficient form without changing their value. This process often involves combining like terms, applying exponent rules, and sometimes considering the properties of operations such as addition, subtraction, multiplication, and division of polynomials and radical expressions.In our exercise example:

• Start by converting any radicals to rational exponents, as shown in earlier steps.
• Multiply the coefficients separately from the variables to streamline calculations.
• Use algebraic properties like the product of powers property to combine terms effectively.

The outcome, \( 10x^{7/12} \), is fully simplified. Such streamlined expressions are not only easier to work with but also enhance clarity when solving equations or performing additional algebraic manipulations. Remember, simplifying is about making expressions easier to handle while maintaining equivalency.