Problem 85 - College Algebra

Question

In Exercises \(85-94,\) simplify using properties of exponents. $$\left(7 x^{1 / 3}\right)\left(2 x^{1 / 4}\right)$$

Step-by-Step Solution

Answer

The simplified expression of \( \left(7 x^{1 / 3}\right)\left(2 x^{1 / 4}\right) \) is \( 14x^{7/12} \).

Step 1: Identify the bases and their respective exponents

In the expression \( \left(7 x^{1 / 3}\right)\left(2 x^{1 / 4}\right) \), there are two bases: the number \(x\) in both parentheses, with respective exponents of \(1/3\) and \(1/4\). The coefficients are 7 and 2.

Step 2: Apply the property of exponents' addition to the same base

When multiplying two expressions with the same base, the exponents are added together. Since both terms contain the base \(x\), their exponents can be added. After summing the exponents, \(1/3 + 1/4 = 7/12\).

Step 3: Multiply the coefficients

The coefficients are 7 and 2, which are being multiplied together. So, \(7 \times 2 = 14\).

Step 4: Combine the coefficients and exponents

Combine the multiplied coefficients and the exponent which was the result of the summation. Write the coefficient first, then the base \(x\), and finally the combined exponent. This gives the simplified expression, \( 14x^{7/12} \)

Key Concepts

Simplifying Expressions

When we simplify mathematical expressions, the goal is to reduce them into their most basic form. This often involves combining like terms and applying arithmetic operations. Simplifying expressions involves a series of steps in which complex parts of an expression are reduced to a more manageable or compact form.

In the expression \( (7x^{1/3})(2x^{1/4}) \), the simplification starts by examining the parts of the expression that can be combined or reduced, which includes identifying similar bases or common factors. Here, the expression has two identical bases, \( x \), allowing us to focus on these to simplify by combining their exponents effectively. This concept is essential in algebra, as it helps make solving equations easier and makes complex expressions less intimidating to work with.

By understanding the rules and properties that apply to different parts of an expression, a student can simplify effectively and efficiently. Simplification ensures clarity and often reveals insights about the expressions or equations being worked on.

Addition of Exponents

Addition of exponents is a crucial rule when working with powers in algebra. It applies when you multiply expressions that have the same base. Remember, only the exponents are added—not the bases.

In our case, the bases of both terms are \( x \). When multiplying \( x^{1/3} \) and \( x^{1/4} \), you simply keep the base and sum the exponents. This follows from the property: \( a^m \times a^n = a^{m+n} \).

Let's compute:

• The exponents are \( \frac{1}{3} \) and \( \frac{1}{4} \).
• Convert these fractions to have a common denominator. Here, \( \frac{1}{3} = \frac{4}{12} \) and \( \frac{1}{4} = \frac{3}{12} \).
• Add them to get \( \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \).

In the product of the terms, the base remains \( x \), and its new exponent is \( \frac{7}{12} \). This is the simplified form of the exponents’ addition seen in the expression.

Multiplying Coefficients

Multiplying coefficients is straightforward arithmetic involving just the numerical values that precede variables. In any algebraic expression, coefficients are the numbers in front of the variable terms and need to be handled just like regular numbers in multiplication.

In the given expression, the coefficients are the numbers 7 and 2. Multiplying these coefficients entails performing simple multiplication as you would with any standard numbers. So, for the expression \( (7x^{1/3})(2x^{1/4}) \), you multiply:

• 7 and 2, which gives you 14.

There are no complicated steps or additional operations required—it is one of the basic algebra tasks involving multiplication. When the coefficients are combined with the result from the addition of exponents, they form the complete simplified expression. Multiplying coefficients helps maintain the accuracy and completeness of the simplification process of algebraic expressions.