/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 142 Simplify each expression. $$ ... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 142

Simplify each expression. $$ \left(2 m n^{4}\right)^{5} $$

Short Answer

Expert verified
The simplified expression is \(32m^{5}n^{20}\).

Step by step solution

01

Identify the Exponents

The given expression is \(\left(2mn^{4}\right)^{5}\). Notice that both the constant \(2\) and the variables \(mn^{4}\) are raised to the power of \(5\).
02

Apply the Power of a Power Property

Take the power of \(5\) and apply it to each component inside the parentheses separately. This property states that \(\left(ab\right)^{n} = a^{n}b^{n}\). Therefore: \((2mn^{4})^{5} = 2^{5} \cdot (m)^{5} \cdot (n^{4})^{5}\)
03

Simplify Each Component

Simplify \(2^{5}\), \(m^{5}\), and \(n^{4\cdot5}\). We get: \2^{5}=32,\ \m^{5}\ is already simplified, and \(n^{4\cdot5}=n^{20}\). Thus the expression becomes: \(32m^{5}n^{20}\)

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Key Concepts

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

Exponents
Understanding exponents is essential in algebra. An exponent indicates how many times a number (the base) is multiplied by itself. For example, in the expression \(2^5\), the base is \(2\) and the exponent is \(5\). This means we multiply \(2\) by itself \(5\) times: 2 脳 2 脳 2 脳 2 脳 2 = 32. Exponents are not just limited to numbers. They can also apply to variables. Consider \(m^5\); here, the variable \(m\) is multiplied by itself \(5\) times. Whether dealing with constants or variables, the rules for exponents remain the same. Breaking the problem down step-by-step can simplify even the most complex expressions.
Power of a Power Property
The 'power of a power' property is a crucial rule in algebra that simplifies expressions with exponents. This property states that \((a^b)^c = a^{b \times c}\). In simple terms, if you have a number or variable raised to an exponent, and then that result is raised to another exponent, you can multiply the exponents together. Let's consider the problem:\[(2mn^4)^5\] According to the power of a power rule, we can distribute the outside exponent to each base inside the parentheses individually:\[2^5(m^5)(n^{4\bullet 5}).\]By adopting this property, we simplify the process significantly, making it easier to manage larger expressions with multiple exponents.
Algebraic Simplification
Algebraic simplification involves reducing an expression to its simplest form. By applying known properties and rules, we make our expressions easier to work with. Simplifying algebraic expressions involves combining like terms and reducing complex expressions step-by-step. Take the simplified components from our problem: \[2^5(m^5)(n^{20}).\] First, simplify each component: * Evaluate \(2^5\) as \(32\).* \(m^5\) is already in its simplest form. * For \(n^{20}\), observe the exponent has been correctly simplified from multiplying \(4\) by \(5\) within the original parenthesis: \[n^{4 \times 5} = n^{20}.\]Putting it all together, the simplified expression is:\[32m^5n^{20}.\]

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