Problem 7 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \left(n^{3}\right)^{3}\left(n^{-3}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Apply the Power Rule

The power rule for exponents states that when raising a power to another power, you multiply the exponents: \[ (a^m)^n = a^{m \cdot n} \]Apply this rule to \((n^3)^3\):\[ (n^3)^3 = n^{3 \cdot 3} = n^9 \]And to \((n^{-3})^3\):\[ (n^{-3})^3 = n^{-3 \cdot 3} = n^{-9} \].

Step 2: Combine Like Bases

When multiplying expressions with the same base, you add their exponents: \[ a^m \cdot a^n = a^{m+n} \]Combine the results from Step 1:\[ n^9 \cdot n^{-9} = n^{9 + (-9)} = n^0 \].

Step 3: Simplify with Zero Exponent

Any non-zero number raised to the power of zero equals one:\[ n^0 = 1 \].Therefore, the expression simplifies to 1.

Key Concepts

Power Rule

The Power Rule is a crucial exponent rule that simplifies expressions where a power is raised to another power. When using the Power Rule, you multiply the exponents together. This can be expressed using the formula:\[ (a^m)^n = a^{m \cdot n} \] For instance, consider the expression \((n^3)^3\). According to the Power Rule, you multiply the exponents 3 and 3:\[ (n^3)^3 = n^{3 \cdot 3} = n^9 \] This simplifies the expression by applying a single exponent to the base. Remember, this rule is specific to exponentiation where you have nested exponentials.

Zero Exponent Rule

Understanding the Zero Exponent Rule is fundamental in the world of exponents. The rule states that any number, except zero, raised to the power of zero, is equal to one. Mathematically, it is represented as:\[ a^0 = 1 \] Why does this happen? It's all about maintaining consistent patterns in arithmetic. When we diminish an exponent step-by-step by dividing by the base, we eventually reach the zero exponent state, which standardizes to one. In this exercise, after applying the power and combining exponents, we ended up with \(n^0\), which simplifies to 1:\[ n^0 = 1 \] Be cautious, this rule does not apply to zero itself since \(0^0\) is undefined.

Combining Exponents

Combining exponents helps to simplify expressions that have the same base. The rule states that when multiplying like bases, you add their exponents:\[ a^m \cdot a^n = a^{m+n} \] In this specific exercise, after applying the power rule, we had two terms: \(n^9\) and \(n^{-9}\). Both these expressions share the same base, 'n'. Thus, by adding the exponents, we get:\[ n^9 \cdot n^{-9} = n^{9+(-9)} = n^0 \] This leads us back to using the Zero Exponent Rule such that \(n^0 = 1\). Combining exponents is a powerful tool in reducing complex expressions into much simpler forms.