Problem 46 - Beginning and Intermediate Algebra

Question

Simplify. $$\left(4 n^{-1}\right)^{2}\left(2 n^{3}\right)^{3}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is: \(128 \cdot n^7\).

Step 1: Distribute the exponent

Apply the power rule, which states that \((ab)^n = a^n b^n\), to distribute the exponent to each factor inside the parenthesis. \[\left(4 n^{-1}\right)^{2}\left(2 n^{3}\right)^{3} = (4^2 \cdot n^{-1\cdot2})(2^3 \cdot n^{3\cdot3})\]

Step 2: Simplify individual terms

Calculate the individual terms by raising each factor to the power. \[ (4^2 \cdot n^{-1\cdot2})(2^3 \cdot n^{3\cdot3}) = (16 \cdot n^{-2})(8 \cdot n^9)\]

Step 3: Multiply the constants and terms with same base

Multiply the constants (16 and 8) together, and then apply the product rule for exponents, which states that \(a^m \cdot a^n = a^{m+n}\), to multiply the terms with the same base (n in this case). \[ (16 \cdot n^{-2})(8 \cdot n^9) = 16\cdot 8 \cdot n^{-2+9}\]

Step 4: Simplify the expression

Complete the multiplications and add the exponents of the terms with the same base. \[16\cdot 8 \cdot n^{-2+9} = 128 \cdot n^7\] After following these steps, the simplified form of the given expression is: \[\left(4 n^{-1}\right)^{2}\left(2 n^{3}\right)^{3} = 128 \cdot n^7\]

Key Concepts

Exponent Rules

Exponent rules are essential tools for dealing with powers in algebraic expressions. These rules simplify expressions that involve numbers or variables raised to a power. Here are some key exponent rules that are commonly used:

• Product of Powers: When multiplying like bases, add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Power of a Power: When raising an exponent to another power, multiply the exponents. For instance, \((a^m)^n = a^{m\times n}\).
• Negative Exponent: A negative exponent represents the reciprocal of the positive exponent. In other words, \(a^{-n} = \frac{1}{a^n}\).
• Zero Exponent: Any number raised to the zero power is equal to one: \(a^0 = 1\), where \(a eq 0\).

Understanding these rules helps in breaking down and simplifying complex expressions, making calculations more manageable.

Simplifying Expressions

Simplifying expressions involves performing operations and applying algebraic rules to make an expression easier to work with or understand. The process reduces expressions to their simplest form by combining like terms and applying mathematical properties:

• Look for terms that can be combined or simplified using mathematical operations.
• Apply exponent rules to simplify terms with exponents.
• Simplify fractions whenever possible to reduce expression size.
• Remove parentheses by distributing or applying the order of operations.

By consistently applying these techniques, mathematicians and students can achieve a tidier, less cluttered expression, which often reveals patterns or solutions that are not immediately visible.

Product Rule for Exponents

The product rule for exponents is a fundamental rule in algebra that facilitates the handling of terms with exponents when multiplying. According to this rule, when multiplying two terms with the same base, you can simply add their exponents. This rule is expressed as:

• \(a^m \times a^n = a^{m+n}\)

It simplifies complex expressions by reducing the number of exponentiated terms. For example, when faced with \(x^2 \times x^3\), rather than working through each term individually, you can directly add the exponents to get \(x^{2+3} = x^5\). The product rule's utility extends to more intricate algebraic expressions like the one in the original exercise \((16 \cdot n^{-2})(8 \cdot n^9)\), simplifying the exponents into a single power of \(n\): \(n^{-2+9} = n^7\). This makes dealing with large, multi-term products much more manageable.