Question

Simplify \(6^5\left(6^2\right)^3\) A. \(6^{11}\) B. \(6^{10}\) C. \(36^{10}\) D. \(36^{11}\)

Short answer

A. \[6^{11}\]

Step-by-step solution

- Understand the Problem

The task is to simplify the expression involving exponents: \[6^5(6^2)^3\]

- Apply the Power Rule

Use the power rule of exponents, which states \[\forall a eq 0, (a^m)^n = a^{mn}\]. So, apply this to \[(6^2)^3\]: \[(6^2)^3 = 6^{2 \times 3} = 6^6\]

- Combine the Exponents

Combine the exponents using the product rule, which states for any non-zero base a: \[a^m \times a^n = a^{m+n} \]. So, combine \[6^5\] and \[6^6\]: \[6^5 \times 6^6 = 6^{5+6} = 6^{11}\]

- Select the Correct Answer

The simplified expression is \[6^{11}\]. So, the correct answer is A. \[6^{11}\].

Key concepts

Power Rule

The power rule is an essential concept when working with exponents. It describes how to handle situations where an exponent is raised to another exponent. The rule states that for any non-zero number \(a\), and any integers \(m\) and \(n\):

\[ (a^m)^n = a^{mn} \].

This means you multiply the exponents together. For example, in the exercise, we had the expression \( (6^2)^3 \). By applying the power rule, we get:

\( (6^2)^3 = 6^{2 \times 3} = 6^6 \).

This rule simplifies more complex exponent expressions by consolidating them into a single power. It is especially useful in simplifying nested exponentials.

Product Rule

The product rule helps us combine exponents with the same base. It's a crucial tool that states:

\[ a^m \times a^n = a^{m+n} \].

This means that when you multiply two exponents with the same base, you simply add the exponents together. For example, in the given problem, after applying the power rule, we had:

\( 6^5 \times 6^6 \).

By using the product rule, we combine these exponents:

\( 6^5 \times 6^6 = 6^{5+6} = 6^{11} \).

This simplifies multiple terms into a single term. It's a handy rule for combining exponential terms during simplification.

Exponents

Exponents are a shorthand way to express repeated multiplication of a number by itself. For example, \( 6^5 \) means \( 6 \times 6 \times 6 \times 6 \times 6 \).

Here are some key points to remember:

• An exponent tells us how many times a number (the base) is multiplied by itself.
• Common rules include the power rule and the product rule, which simplify complex expressions.
• Exponents follow certain properties that make it easier to handle larger numbers.

Understanding these basic properties can simplify working with exponents in algebra and higher math. They form the foundation for more advanced topics, such as logarithms and polynomial functions.