Problem 60 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(c^{-9} \cdot c^{14}\)

Step-by-Step Solution

Answer

The simplified form is \(c^5\).

Step 1: Apply the Product of Powers Property

When multiplying two powers with the same base, you can add the exponents. Here, both terms have the base \(c\). Apply the rule: \(a^m \cdot a^n = a^{m+n}\). This gives us \(c^{-9} \cdot c^{14} = c^{-9 + 14}\).

Step 2: Simplify the Exponent

Now simplify the exponent by performing the addition: \(-9 + 14 = 5\). Therefore, \(c^{-9 + 14} = c^5\). This is already in its simplest form with no negative exponents.

Key Concepts

Product of Powers Property

When dealing with exponents, the "product of powers" property is a very handy rule to simplify expressions. This property states that when you multiply two powers with the same base, you simply add their exponents together.
For instance, consider the expression \(a^m \cdot a^n\). According to the product of powers property, this equals \(a^{m+n}\).
This rule is helpful because it simplifies the multiplication of numbers in exponential form, making calculations less cumbersome.
In the original exercise with \(c^{-9} \cdot c^{14}\), both terms share the same base, \(c\). By applying this property, you get \(c^{-9+14}\). This step smartly condenses the multiplication into a single power, \(c^5\).
Using the product of powers property simplifies complex algebraic expressions, thus making your calculations quicker and easier.

Exponents

Exponents, sometimes called powers, are a shorthand for repeated multiplication of the same number. An expression with an exponent, like \(a^n\), means that you multiply the base \(a\) by itself \(n\) times.
Exponents are powerful tools in math because they allow us to express very large or very small numbers succinctly.
Consider \(c^5\) in the solution, which translates to multiplying \(c\) by itself five times: \(c \cdot c \cdot c \cdot c \cdot c\).
This compact form eases lengthy calculations and highlights patterns and relationships in mathematical expressions. As you handle exponents more, you’ll begin to see their applicability in simplifying and solving equations.
Whether in mathematics or science, exponents help encapsulate complex operations into more manageable notations.

Negative Exponents

Negative exponents can seem daunting at first, but they are quite intuitive once you understand them. A negative exponent represents the reciprocal of the base raised to the opposite positive exponent.
For example, \(a^{-n}\) equals \(\frac{1}{a^n}\). This implies that a negative exponent flips the fraction, essentially moving the base from the numerator to the denominator or vice versa.
In the original problem, \(c^{-9}\) and \(c^{14}\) are multiplied using the product of powers property. The resulting expression, \(c^{-9+14}\), simplifies to \(c^5\), eliminating the negative exponent characteristic.
This step demonstrates how manipulating negative exponents through established rules can simplify expressions, making them easier to interpret and solve.

• Negative exponents indicate division.
• Use reciprocal to convert them to positive exponents.

Recognizing and applying these principles adequately will minimize confusion and enhance proficiency with exponential expressions.