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Chapter 1: Problem 97

Simplifying \(\left(a^{m}\right)^{n}\) $$ \left(9^{3}\right)^{-4} $$

Short Answer

Expert verified
The simplified form is \(9^{-12} = \frac{1}{9^{12}}\).

Step by step solution

01

Identify the General Rule

Recall the exponent rule: \[\big(a^{m}\big)^{n} = a^{m \times n} \]. This states that when raising a power to another power, you multiply the exponents.
02

Apply the Rule to the Given Expression

Use the rule to simplify the expression \(\big(9^{3}\big)^{-4}\). Here, \(a = 9\), \(m = 3\), and \(n = -4\). So \((9^{3})^{-4} = 9^{3 \times (-4)} = 9^{-12}\).
03

Simplify the Final Expression

Rewrite the simplified expression with a positive exponent if needed. \(9^{-12}\) can be written as \(\frac{1}{9^{12}}\).

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Key Concepts

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

Power Rule
To simplify complex exponent expressions, it's crucial to understand a few foundational rules. One of the main rules is the Power Rule. This rule states that when you have an exponent raised to another exponent, you multiply them. The general form is \((a^m)^n = a^{m \times n}\). For example, in the expression \((9^3)^{-4}\), we use the Power Rule to simplify it. Here, we have \(9^3\) raised to \(-4\), which simplifies to \((9^3)^{-4} = 9^{3 \times -4} = 9^{-12}\). This rule helps in making complex expressions easier to manage.
Negative Exponents
Negative exponents can be tricky, but they follow a simple rule. A negative exponent indicates that the base should be taken as the reciprocal. For instance, \(a^{-n} = \frac{1}{a^n}\). So, if we take our simplified expression \(9^{-12}\), we can rewrite it with a positive exponent by flipping it to the reciprocal form: \(9^{-12} = \frac{1}{9^{12}}\). This form is often more manageable and easier to understand. Always remember, a negative exponent simply means 'take the reciprocal.'
Exponent Multiplication
The rules for multiplying exponents come in handy when dealing with expressions that involve raising a power to another power. Another key aspect is that if you multiply numbers with the same base, you add the exponents. For example, \(a^m \times a^n = a^{m+n}\). However, in the given problem, we focus on the Power Rule for exponent multiplication, which simplifies \((a^m)^n\). Combining these rules helps in simplifying and solving complex exponent expressions efficiently.

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