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Chapter 5: Problem 10

Apply the product rule for exponents, if possible, in each case. $$ 11^{6} \cdot 11^{4} $$

Short Answer

Expert verified
\(11^{10}\)

Step by step solution

01

Identify the Base and Exponents

In the expression \(11^{6} \cdot 11^{4}\), identify the common base (11) and the respective exponents (6 and 4).
02

Apply the Product Rule for Exponents

The product rule for exponents states that when multiplying two expressions with the same base, you add the exponents. Therefore, apply the rule: \(11^{6} \cdot 11^{4} = 11^{6+4}\).
03

Simplify the Expression

Add the exponents together: \(6 + 4 = 10\). Thus, the expression simplifies to \(11^{10}\).

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Key Concepts

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

Base and Exponents
When dealing with exponents, it is crucial to understand the structure of expressions like the one in our example, \(11^6 \cdot 11^4\).
  • The base is the number that is being multiplied by itself.
  • The exponent indicates how many times the base is used as a factor.

In our example, the base is 11 in both terms, and the exponents are 6 and 4 respectively. This fundamental understanding is essential before applying any rules to simplify the expression.
Simplifying Expressions
Simplifying expressions is essential for handling complex mathematical problems more easily.
  • Look for common elements in the expression, such as a shared base.

  • Combine like terms using rules of exponents and algebra.

For example, in \(11^6 \cdot 11^4\), both terms share the base 11. This allows us to use the product rule to simplify our expression efficiently. Breaking the problem into smaller, manageable steps, as shown in the solution process, is a key technique in problem-solving.
Exponent Addition
One of the core principles you often use with exponents is the product rule, which involves exponent addition.

The product rule for exponents states: When multiplying two powers that have the same base, simply add the exponents together.

In our case: $$ 11^6 \cdot 11^4 = 11^{6+4} = 11^{10} $$
  • Identify the common base.

  • Add the exponents.

Adding exponents is a powerful tool that simplifies complex expressions into more manageable forms, helping to solve problems more quickly and accurately.

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Most popular questions from this chapter

Evaluate. Express answers in standard notation. $$ \frac{3 \times 10^{-2}}{12 \times 10^{3}} $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ 3^{5} \cdot 3^{-6} $$

Evaluate. Express answers in standard notation. $$ \frac{16 \times 10^{5}}{4 \times 10^{8}} $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \left(\frac{8 x^{-6} y^{3}}{x^{4} y^{-4}}\right)^{-2} $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \frac{(3 p q) q^{2}}{6 p^{2} q^{4}} $$
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