Problem 60 - Algebra for College Students

Question

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(3 x^{\frac{1}{4}}\right)\left(5 x^{\frac{1}{3}}\right)\)

Step-by-Step Solution

Answer

15x^{\frac{7}{12}}

Step 1: Multiply the Constants

Start by multiplying the coefficients outside of the variables. In this case, multiply 3 by 5 to get 15.

Step 2: Add the Exponents for the x Terms

Since the base of both terms is the same (x), we can add the exponents. Add the exponents \(\frac{1}{4}\) and \(\frac{1}{3}\). To do this, find a common denominator, which is 12. Convert \(\frac{1}{4}\) to \(\frac{3}{12}\) and \(\frac{1}{3}\) to \(\frac{4}{12}\). Now, add them: \(\frac{3}{12} + \frac{4}{12} = \frac{7}{12}\).

Step 3: Write the Result

Combine the result from Step 1 and Step 2 to express the product as \(15x^{\frac{7}{12}}\). Make sure to use positive exponents in the final expression.

Key Concepts

Exponents

Exponents are a way to express repeated multiplication. If you see something like \( x^a \), it tells you that \( x \) is multiplied by itself \( a \) times. In algebra, exponents can be whole numbers, fractions, or even zero.
When multiplying terms with the same base, you simply add the exponents. For example, \( x^{rac{1}{4}} \) and \( x^{rac{1}{3}} \) have the same base \( x \), so their exponents add up: \( \frac{1}{4} + \frac{1}{3} \).
Finding the common denominator helps in adding fractions, making it easier to arrive at the simplified exponent.

Multiplication

Multiplication in algebra involves multiplying coefficients and variables. Coefficients are the numbers in front of the variables. For instance, in \( 3x \), the number 3 is a coefficient.
When multiplying terms, start by multiplying these coefficients. In the exercise, multiplying 3 and 5 results in 15.
If variables are part of the terms, follow up by applying exponent rules to them.

• Multiply coefficients separately.
• Add exponents if the variable bases are the same.
• Apply multiplication consistently across terms.

This step-by-step approach ensures you get the correct simplified form.

Variables

Variables in algebra are symbols, usually letters, that represent numbers. They allow you to create general formulas and expressions without assigning specific values.
In our exercise, the variable \( x \) appears with different exponents and is an essential part of the expression.
When the variables have the same base, as they do here, you manage their exponents by adding them when multiplying terms.

• Recognize the common variables in each term.
• Apply exponent rules to these variables efficiently.
• Maintain variables throughout calculations for accuracy.

Proper handling of variables is crucial for simplification.

Algebraic Expressions

Algebraic expressions combine numbers, variables, and operators (like +, -, *, /) to represent mathematical ideas. They're the backbone of algebra.
Simplifying an algebraic expression involves reducing it to its simplest form while maintaining its value. In the multiplication context, as shown in the exercise, this process requires combining like terms.

• Identify the basis and structure of each term.
• Apply operations systematically.
• Express the final result using only positive exponents.

Simplification enhances clarity and makes expressions more manageable, aiding further problem-solving.