Problem 35 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. \(\left(c^{8}\right)^{10}\)

Step-by-Step Solution

Answer

\(\left(c^{8}\right)^{10}=c^{80}\)

Step 1: Identify the Base and Exponents

In the given expression \(\left(c^{8}\right)^{10}\), \(c\) is the base and \(8\) and \(10\) are the exponents.

Step 2: Use the Rule of Exponents

According to the rule of exponent \((a^m)^n = a^{m*n}\), where \(a\) is the base, \(m\) and \(n\) are the exponents. Our base is \(c\) and the exponents are \(8\) and \(10\). Applying the rule, we get \(c^{8*10}\).

Step 3: Simplify the Expression

To finish the problem, we need to complete the multiplication. We simply multiply \(8*10\) to get \(80\). Thus, \(\left(c^{8}\right)^{10}=c^{80}\).

Key Concepts

Understanding Powers

Powers are a fundamental concept in mathematics that denote repeated multiplication of a number by itself. When we talk about powers, we refer to the expression where a number, known as the "base," is raised to a certain value called the "exponent." The exponent indicates how many times the base is multiplied by itself. For example, in the expression \(c^8\), \(c\) is the base and \(8\) is the power, meaning \(c\) is multiplied by itself 8 times.

• The base is the number being repeatedly multiplied.
• The power or exponent signifies the number of times the base is used in the multiplication process.

Understanding powers is crucial as they simplify the expression of large numbers. By using powers, you can express significantly large values in a compact form.

Base and Exponents

In mathematical terms, the base and exponent play pivotal roles in forming the expression of a power. Think of the base as the "root" number from which the operation originates, while the exponent tells you "how far to go" with that base.

• The base remains constant in expressions or calculations unless further modified by external operations.
• Exponents or powers provide the degree of repeated multiplication.

For example, in the expression \((c^8)^{10}\), \(c\) is the base. First, \(c\) is raised to the 8th power. If an additional exponent (in this case, 10) is applied outside the brackets, it indicates that the initial power itself is being multiplied repeatedly. This results in a more complex calculation that can be simplified using the rule of exponents.

The Rule of Exponents

The rule of exponents is a set of guidelines that help simplify expressions involving powers. One of the key rules states that when terms like \((a^m)^n\) are encountered, they can be simplified to \(a^{m\cdot n}\). This rule makes calculations much easier, particularly when dealing with multiple layers of exponents.In the original exercise, you have \((c^8)^{10}\). To resolve this, apply the rule \((a^m)^n = a^{m\cdot n}\) by substituting \(c\) for \(a\), \(8\) for \(m\), and \(10\) for \(n\). The expression simplifies to \(c^{8\cdot 10} = c^{80}\).

• This simplification saves time compared to multiplying the base by itself successive times.
• Understanding and applying exponential rules are essential for tackling more complex mathematical problems.

Using these rules correctly ensures you're efficiently simplifying and solving power expressions, avoiding tedious multiplication.