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Chapter 2: Problem 78

Simplify the expression using one of the power rules. $$\left(-4^{5}\right)^{3}$$

Short Answer

Expert verified
The simplified expression is \((-4)^{15}\) using the power rule \((a^m)^n = a^{m\cdot n}\).

Step by step solution

01

Identify the base and exponents

In this expression, we have the base -4, raised to the power of 5, which is then raised to another exponent, 3. So, the base is -4, and the exponents are 5 and 3.
02

Apply the power rule

We can now apply the power rule for exponents: \((a^m)^n = a^{m\cdot n}\). In our case, this becomes: \[\left((-4)^{5}\right)^{3} = (-4)^{5\cdot 3}\]
03

Multiply the exponents

Now multiply the exponents 5 and 3: \[(-4)^{5\cdot 3} = (-4)^{15}\]
04

Simplify the expression

Finally, the simplified expression is: \((-4)^{15}\) This is the final simplified expression using the power rules.

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Key Concepts

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

Power Rule
When dealing with exponents, the power rule is a very important tool. It helps simplify expressions like \((a^m)^n\) by allowing us to multiply exponents.
  • This rule is applied when an exponential expression is raised to another power.
  • The power rule states that \( (a^m)^n = a^{m \cdot n} \).
Understanding and applying this rule correctly can make simplifying complex expressions much easier.
Think of it as a shortcut that combines two steps into one, reducing the potential for errors and saving time. In our exercise, \( ig((-4)^5\big)^3 \) becomes \(-4^{15}\) by applying the power rule.
Simplifying Expressions
Simplifying an expression means rewriting it in its simplest or most efficient form. For expressions with exponents, this often involves using the power rule or other exponent rules.
  • First, identify the base and the powers involved in the expression.
  • Next, apply the relevant exponent rules like the power rule to reduce the expression.
  • The result should be a single term or a less complex version of the expression.
In our example, we identified \((-4)^{5}\) raised to the power of \(3\). By using the power rule, we were able to combine the exponents efficiently, simplifying the expression to \((-4)^{15}\). This process removes unnecessary complexity.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations. They can include addition, subtraction, multiplication, division, and exponents.
  • A critical skill in algebra is manipulating these expressions to solve equations or simplify problems.
  • When dealing with exponents, the rules of algebra help us combine and reduce terms.
  • This requires understanding the order of operations, properties of exponents, and algebraic functions.
In the exercise, \((-4)^{5}\) raised to the third power represents an algebraic expression that can be simplified by following algebraic rules. By simplifying, we transform it into a form that is easier to work with, such as \((-4)^{15}\). Mastering these concepts allows you to solve more complex mathematical problems efficiently.

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Most popular questions from this chapter

Express each number in scientific notation, then solve the problem. In Wisconsin in 2001 , approximately \(1,300,000\) cows produced \(2.21 \times 10^{10}\) lb of milk. On average, how much milk did each cow produce?

Write each number in scientific notation. -145,000

Simplify the expression using the product rule. Leave your answer in exponential form. $$\left(\frac{5}{3} x^{2}\right)(12 x)\left(-2 x^{3}\right)$$

Simplify the expression using the product rule. Leave your answer in exponential form. $$(-3) \cdot(-3)^{5} \cdot(-3)^{2}$$

Perform the operation as indicated. Write the final answer without an exponent. \(\frac{-3 \times 10^{5}}{6 \times 10^{8}}\)
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