Problem 137 Simplify each expression. $$ ... [FREE SOLUTION]

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Chapter 5: Problem 137

Simplify each expression. $$ \left(a^{2}\right)^{6} \cdot\left(a^{3}\right)^{8} $$

Short Answer

Expert verified

$a^{36}$

Step by step solution

- Apply Power Rule to First Term

Simplify $(a^2)^6$ using the power rule $(a^m)^n = a^{mn}$. This gives us $a^{2 \times 6} = a^{12}$.

- Apply Power Rule to Second Term

Next, simplify $(a^3)^8$ using the same power rule $(a^m)^n = a^{mn}$. This gives us $a^{3 \times 8} = a^{24}$.

- Multiply the Simplified Terms

Now multiply the simplified terms $a^{12} \cdot a^{24}$ using the product rule $a^m \cdot a^n = a^{m+n}$. This results in $a^{12+24} = a^{36}$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule

When simplifying algebraic expressions with exponents, the power rule is an essential concept to master. The power rule states that when you have a power raised to another power, you multiply the exponents. For example, $ (a^m)^n = a^{mn} $.
In this exercise, we used the power rule twice:

First, simplify $ (a^2)^6 = a^{2 \times 6} = a^{12} $.
Then, simplify $ (a^3)^8 = a^{3 \times 8} = a^{24} $.

Mastering this rule allows you to handle more complex expressions with nested exponents effectively.

It's helpful to practice this rule in various contexts to feel comfortable using it in different problems.

Product Rule

The product rule is another important rule when working with exponents. This rule states that when multiplying two expressions with the same base, you add their exponents. For example, $ a^m \cdot \ a^n = a^{m+n} $.
In this problem, after applying the power rule to each term, we multiplied the resulting expressions:

Combine $ a^{12} \cdot \ a^{24} $
Add the exponents: $ 12 + 24 = 36 $
Result is $ a^{36} $

This rule is crucial for simplifying expressions and solving equations.

Knowing when and how to use the product rule will save you a lot of time and reduce errors in your calculations.

Exponents

Exponents are a key element of algebra that indicate how many times a number (the base) is multiplied by itself. For example, $ a^2 $ means $ a \cdot \ a $. Understanding how to manipulate exponents through rules like the power rule and product rule makes solving algebraic expressions much more manageable.
Let's recap the exercise:

Simplify $ (a^2)^6 $ to get $ a^{12} $
Simplify $ (a^3)^8 $ to get $ a^{24} $
Combine the results using the product rule to get $ a^{36} $

These foundational skills in working with exponents are vital across many areas of math.

From simplifying expressions to solving higher-level problems, mastering these rules provides a strong mathematical foundation.

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91影视

Simplify each expression. $$ \left(a^{2}\right)^{6} \cdot\left(a^{3}\right)^{8} $$

Short Answer

Step by step solution

- Apply Power Rule to First Term

- Apply Power Rule to Second Term

- Multiply the Simplified Terms

Key Concepts

Power Rule

Product Rule

Exponents

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