Problem 37 - Beginning and Intermediate Algebra

Question

Simplify. Assume all variables represent nonzero real numbers. The answer should not contain negative exponents. $$5\left(2 m^{4} n^{7}\right)^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(20m^8n^{14}\).

Step 1: Apply the exponentiation rule to the parentheses involving the power of 2

First off, let's remember that when raising a product to a power, one must raise each factor inside the parentheses to that power: \[((ab)^n = a^nb^n]\] In this case, apply the exponent 2 to all of the factors inside the parentheses (2, \(m^4\), \(n^7\)): \[5(2m^4n^7)^2 = 5(2^2(m^4)^2(n^7)^2)\]

Step 2: Simplify the exponents inside the parentheses

Now, let's simplify the exponents inside the parentheses. Remember that when raising a power to another power, we must multiply the exponents: \[(a^m)^n = a^{mn}\] Now let's apply this rule to our expression: \[5(2^2(m^4)^2(n^7)^2) = 5(4m^{4\cdot 2}n^{7\cdot 2}) = 5(4m^8n^{14})\]

Step 3: Distribute the constant outside the parentheses

Finally, distribute the constant 5 to all terms inside the parentheses: \[5(4m^8n^{14}) = (5\cdot4)m^8n^{14} = 20m^8n^{14}\] So, the simplified expression is: \[20m^8n^{14}\]

Key Concepts

Exponentiation Rules

Exponentiation rules are essential tools for simplifying expressions involving powers. When you encounter an expression like \((ab)^n\), the rule is to apply the exponent to each factor separately: \(a^n b^n\). This helps break down complex expressions into manageable parts.

• For example, in \((2m^4n^7)^2\), you need to apply the power of 2 to 2, \(m^4\), and \(n^7\) separately.
• This results in \(2^2 (m^4)^2 (n^7)^2\).

Remember that exponents must be handled carefully to ensure proper simplification. This approach reduces errors and streamlines the process.

Distributive Property in Algebra

The distributive property is crucial in algebraic simplification, especially when dealing with expressions enclosed in parentheses. It allows you to multiply a single term by each term inside the parentheses individually. For the expression \(5(4m^8n^{14})\), you use this property to distribute the 5.

• Multiply 5 by each term inside: \((5\cdot 4) m^8 n^{14} = 20 m^8 n^{14}\).
• This property helps in distributing and simplifying expressions efficiently.

Applying this property ensures every component of an expression is accounted for and simplifies calculations by breaking them into smaller, more straightforward steps.

Negative Exponents Elimination

Eliminating negative exponents is vital for simplifying expressions to their standard form. Negative exponents imply reciprocals, so transforming them into positive exponents makes calculations more intuitive and expressions easier to manage.

• If you encounter an expression like \(a^{-n}\), it becomes \(\frac{1}{a^n}\).
• In our problem, if a negative exponent were present, converting it to a positive exponent would be essential for proper simplification.

Ensuring all exponents are positive not only simplifies the expression but matches the conventional format expected in final answers.