Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \left(-6 x^{7 / 5}\right)\left(2 x^{8 / 5}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-12 x^3\).

Step 1: Distribute Constants

First, we multiply the constants outside of the variables. In this expression, these constants are -6 and 2. Multiply them together: \[ -6 \times 2 = -12 \]

Step 2: Apply the Power of a Product Rule

Next, apply the power of a product rule, which states that when multiplying like bases, you simply add their exponents. The base here is \(x\), and we add the exponents: \(\frac{7}{5} + \frac{8}{5}\).\[\left(-6 x^{7 / 5}\right)\left(2 x^{8 / 5}\right) = -12 x^{(7 / 5 + 8 / 5)}\]

Step 3: Simplify the Exponents

Now we simplify the exponent by adding the fractions together parts: \[\frac{7}{5} + \frac{8}{5} = \frac{15}{5} = 3\]Thus, the expression becomes \(-12 x^3\).

Key Concepts

Product Rule

The product rule is a crucial concept when dealing with algebraic expressions, especially when multiplying terms that have the same base. This rule tells us what to do when you multiply two powers that have the same base.

• The base is the number that is raised to a power.
• The exponent is the power to which the base is raised.
• When you multiply terms with the same base, just add the exponents together.

For example, in the expression \( x^a \times x^b \), the product rule states that you simply add the exponents: \( x^{a+b} \). This simplifies computations by combining the powers into one single term with the added exponent. Understanding the product rule makes it easier to handle expressions involving powers and exponents.

Simplifying Expressions

Simplifying expressions in algebra means making them as compact and manageable as possible. This often involves combining like terms and applying various algebraic rules, such as the product rule.

• Start by distributing or multiplying any constants.
• Use algebraic rules like combining exponents to simplify further.

In the exercise, simplifying is done by first multiplying the constants, \(-6 \times 2\) to get \-12\, and then simplifying the exponents using the product rule. By adding the exponents \left(\frac{7}{5} + \frac{8}{5}\right)\, we achieve a simplified form of the expression, which avoids any further arithmetic complications.

Exponents

Exponents are a shorthand way to show how many times a number, known as the base, is multiplied by itself. An exponent is written as a small number to the upper right of the base. In algebra, rules for handling exponents guide us in simplifying complex expressions.

• Adding exponents: When multiplying bases with the same number, add their exponents. This is the product rule.
• Subtracting exponents: When dividing bases with the same number, subtract the exponents.

For instance, if you have \(x^{7/5} \times x^{8/5}\), using the product rule, you would add the exponents to get \(x^{15/5}\), which again simplifies to \(x^3\). Getting familiar with these rules is essential for simplifying expressions quickly and accurately.

Fractions

Fractions represent parts of a whole and are often used in algebra to express divisions or proportions. They appear in exponentiation frequently, especially when dealing with non-integer exponents in algebraic expressions.

• Adding fractions: Make sure the denominators are the same, then add the numerators.
• Simplifying fractions: Divide the numerator and the denominator by their greatest common divisor (GCD).

In expressions where exponents are fractions, make sure that before adding them, they have a common denominator. For example, \(\frac{7}{5} + \frac{8}{5} = \frac{15}{5}\), which simplifies to 3. Handling fractions well is a necessary skill for dealing with algebraic simplifications and making calculations smooth and error-free.