Simplify each expression. See Example 1. n⁶ • n⁴ • n

Verified step by step guidance

1. Identify the base and the exponents in the expression \(n^{6} \cdot n^{4} \cdot n\). Here, the base is \(n\) for all terms.
2. Recall the exponent rule for multiplication with the same base: \(a^{m} \cdot a^{n} = a^{m+n}\). This means you add the exponents when multiplying powers with the same base.
3. Rewrite the expression by explicitly showing the exponents: \(n^{6} \cdot n^{4} \cdot n^{1}\) (since \(n\) is the same as \(n^{1}\)).
4. Add the exponents together: \$6 + 4 + 1$ to get the new exponent.
5. Express the simplified form as \(n^{6+4+1}\), which is \(n^{11}\), but do not calculate the sum explicitly here as per instructions.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers of the same base. Specifically, when multiplying terms with the same base, you add their exponents. For example, n^a * n^b = n^(a+b). This rule is essential for simplifying expressions like n⁶ • n⁴ • n.

Base Consistency in Exponentiation

When applying exponent rules, the base must be the same for the rules to hold. In the expression n⁶ • n⁴ • n, all terms share the base 'n', allowing the exponents to be combined. Recognizing consistent bases is crucial before performing operations on exponents.

Simplification of Algebraic Expressions

Simplification involves rewriting expressions in a more compact or standard form without changing their value. For powers, this means combining like terms and reducing the expression to a single term when possible, such as combining n⁶ • n⁴ • n into n^(6+4+1) = n¹¹.