Problem 12 - Elementary Algebra

Question

\(c^{10} c^{3}\)

Step-by-Step Solution

Answer

c^{13}

Step 1: Identify the Base

Both terms in the expression have the same base, which is 'c'.

Step 2: Identify the Exponents

The exponent of the first term is 10, and the exponent of the second term is 3.

Step 3: Apply the Product Rule of Exponents

When multiplying terms with the same base, add their exponents. Therefore, add the exponent 10 to the exponent 3.

Step 4: Calculate the Sum of the Exponents

The sum of the exponents is 10 + 3 = 13.

Step 5: Write the Final Expression

Combine the base and the new exponent: \[ c^{10} \times c^{3} = c^{13} \]

Key Concepts

exponents

Exponents are a way to represent repeated multiplication of a number by itself. In the expression \( c^{10} \), the 10 is the exponent, and it tells us that the base (which is 'c' in this case) is multiplied by itself 10 times. Understanding exponents is crucial in making complex mathematical operations easier.

Key points about exponents:

• Exponents are also called powers.
• The exponent is written as a small number above and to the right of the base.
• They indicate how many times the base is multiplied by itself.

Using exponents simplifies the process of working with large numbers by reducing the need to write out long products.

base

The base is the number that is repeated in multiplication when using exponents. In our example expression \( c^{10} \), 'c' is the base. The base is a fundamental part of the exponentiation process because it is the number that's being repeatedly multiplied.

Understanding the base:

• It's the primary number in the expression.
• It remains the same while the exponent increases or decreases.
• The base can be any number, a constant, or a variable like 'c' in our case.

In complex expressions, keeping track of the base helps in the proper application of exponent rules, like the product rule of exponents.

multiplication of powers

When multiplying powers with the same base, we use the product rule of exponents. This rule states that we add the exponents. For example, in the expression \( c^{10} \times c^{3} \), both terms have the same base 'c'.

Using the product rule of exponents:

• Add the exponents: 10 and 3.
• The sum is 13.
• The simplified expression is \( c^{13} \).

This rule is helpful because it turns a potentially complex multiplication into a single term with a new exponent, making it easier to handle in further calculations.

simplifying expressions

Simplifying expressions involves making them as concise and efficient as possible by applying mathematical rules. In our exercise, we simplified \( c^{10} \times c^{3} \) to \( c^{13} \) using the product rule of exponents.

Steps to simplify expressions with exponents:

• Identify the base and the exponents.
• Apply the appropriate exponent rules, like addition or subtraction of exponents.
• Combine the base with the new exponent.

Simplifying expressions is a fundamental skill in algebra and helps to solve equations more easily and accurately.