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Chapter 12: Problem 118

Simplify. $$ \left(n^{15}\right)^{2} $$

Short Answer

Expert verified
n^{30}

Step by step solution

01

Identify the Power Rule

Recall the power rule of exponents which states that \( \(a^m\) ^n = a^{m \cdot n} \). This rule will help simplify the given expression.
02

Apply the Power Rule

Apply the power rule to the expression \( \left( n^{15} \right)^2 = n^{15 \cdot 2} \).
03

Simplify the Exponent

Multiply the exponents in the expression: \(15 \cdot 2 = 30 \). So \( \left( n^{15} \right)^2 = n^{30} \).

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Key Concepts

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

power rule of exponents
When dealing with exponents, the power rule is one of the most useful tools you can use. This rule states that when you have a power raised to another power, you simply multiply the exponents. Mathematically, it is written as \((a^m)^n = a^{m \times n}\). For example, if you have \((x^3)^2\), you multiply the exponents 3 and 2, resulting in \(x^{6}\). This rule helps make complicated expressions much simpler.
multiplying exponents
Multiplying exponents is straightforward when you use the power rule. You take the base raised to an exponent, then raise it again to another exponent.

For instance, consider the expression \((n^{15})^2\). Here, the base is \(n\), the first exponent is 15, and the second exponent is 2. Using the power rule, you multiply these exponents together: \(15 \times 2 = 30\).
By following this process, the expression simplifies to \(n^{30}\). This method allows you to handle even very large exponents with ease.
simplification
Simplification is the final and crucial step when working with exponents. By reducing expressions to their simplest form, you make them easier to evaluate and understand.
For example, with our initial problem \((n^{15})^2\), you've applied the power rule to get \(n^{30}\).
This process of simplification removes any complexity and makes it clear what the final result is.
  • Always try to simplify expressions to their lowest terms.
  • Use rules like the power rule to aid in simplification.
  • Repeated practice will help you recognize how to simplify complex expressions quickly.

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