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Chapter 3: Problem 36

Rewrite each expression as simply as you can. $$\left(-m^{2} n^{3}\right)^{4}$$

Short Answer

Expert verified
The simplified expression is \({m^8 n^{12}}\).

Step by step solution

01

Distribute the Exponent

First, apply the exponent to each term inside the parentheses. Use the power rule \( (a^m)^n = a^{m\cdot n} \): \[ \left( -m^2 n^3 \right)^4 = (-1)^4 (m^2)^4 (n^3)^4 \]
02

Simplify Each Exponentiation

Next, simplify each of the individual exponentiations: \[ (-1)^4 = 1, \ (m^2)^4 = m^{2\cdot4} = m^8, \ (n^3)^4 = n^{3\cdot4} = n^{12} \]
03

Combine the Results

Combine the simplified terms from Step 2 to get the final result: \[ \left( -m^2 n^3 \right)^4 = 1 \cdot m^8 \cdot n^{12} = m^8 n^{12} \]

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Key Concepts

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

power rule for exponents
The power rule for exponents is a fundamental concept in algebra, especially when simplifying expressions. The rule states that when you raise a power to another power, you multiply the exponents. In mathematical terms, this is written as \( (a^m)^n = a^{m \times n} \).

For example, when you have \( (m^2)^4 \), according to the power rule, you multiply the exponents 2 and 4 together: \( 2 \times 4 = 8 \). So, \( (m^2)^4 = m^8 \).

This rule helps simplify expressions quickly and accurately. Remembering this rule will make it easier to solve complex algebraic equations involving multiple exponents.
distributing exponents
Distributing exponents means applying the exponent outside the parentheses to each term inside the parentheses. This step is essential for simplifying expressions, especially when dealing with multiple variables and constants.

For instance, consider the expression \( (-m^2 n^3)^4 \). Following the distribution rule, we distribute the exponent 4 to each term inside the parentheses:
  • \( (-1)^4 \), since -1 is all alone, and exponentiated values of integer properties are applied to it.
  • \( (m^2)^4 \) 鈥� we apply the exponent 4 to \( m^2 \).
  • \( (n^3)^4 \) 鈥� we apply the exponent 4 to \( n^3 \).
Distributing exponents simplifies the original expression and prepares it for further reduction using algebraic rules. In the example, we get:

\( (-1)^4 (m^2)^4 (n^3)^4 \). As a next step, we simplify each term.
exponentiation
Exponentiation is the process of raising a number or a variable to a power. In our problem, we perform exponentiation on each term. Let's dive into each step:

  • First, we have \( (-1)^4 \). Since the base is -1, raising it to any even number gives us 1: \( (-1)^4 = 1 \).

  • Next, \( (m^2)^4 \). According to the power rule, we multiply the exponents: \( m^{2 \times 4} = m^8 \).

  • Finally, \( (n^3)^4 \). Again, using the power rule: \( n^{3 \times 4} = n^{12} \).
Combining all these, we get: \( 1 \times m^8 \times n^{12} = m^8 n^{12} \).

Understanding exponentiation helps you work with powers more confidently and ensures you follow the correct steps while solving algebraic expressions.

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Most popular questions from this chapter

Prove that each number is rational by finding a pair of integers whose ratio, or quotient, is equal to the number. $$ 3.56 $$

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In Exercises 28鈥�35, find the indicated roots without using a calculator. Challenge the eighth roots of \(x^{16}\)

Rewrite each expression using a single base and a single exponent. $$\left(\frac{m^{84}}{m^{12}}\right)^{x}$$

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