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Chapter 0: Problem 44

Simplify each exponential expression in Exercises 23鈥�64. $$\left(-3 x^{4} y^{6}\right)^{3}$$

Short Answer

Expert verified
The simplified expression of \((-3 x^{4} y^{6})^{3}\) is -27\(x^{12}y^{18}\).

Step by step solution

01

Identify the Base Terms and Powers

Here, we have three base terms: -3, \(x^{4}\), and \(y^{6}\). These are all being raised to the power of 3.
02

Apply the Power Rule

According to the power rule, when a power is raised to another power, powers are multiplied. So, we apply the rule to each base term separately: \((-3)^{3}\), \((x^{4})^{3}\), and \((y^{6})^{3}\).
03

Simplify Each Term

The term \((-3)^{3}\) equals -27, \((x^{4})^{3}\) equals \(x^{12}\) since we multiply 4 by 3, and similarly \((y^{6})^{3}\) becomes \(y^{18}\).

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Key Concepts

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

Understanding the Power Rule
The power rule is a guideline in exponential expressions. It helps us simplify expressions where an exponent is raised to another exponent.

When you see something like \((a^m)^n\), this tells us to multiply the exponents: \((m \times n)\). Applying this concept simplifies the expression to \(a^{m \times n}\).

If you look at the exercise, \((-3 x^{4} y^{6})^{3}\), apply the rule to each part separately:
  • The constant part, \((-3)^3\), which equals -27.
  • The variable \(x\) with \(x^4\) raised to the 3rd power becomes \(x^{4 \times 3} = x^{12}\).
  • The variable \(y\) follows similarly: \(y^{6 \times 3} = y^{18}\).
This rule ensures your calculations remain accurate and straightforward.
Simplifying Exponential Expressions
Simplifying expressions is about making them easier to understand and work with, without changing their value.

After applying the power rule, combine all simplified parts to get the final form.
  • For \((-3)^3\), you calculate the power of the negative integer, resulting in -27.
  • For the variables \(x\) and \(y\), using the power rule means multiplying the exponents.
This gives us direct values for each term. The simplified expression, \(-27x^{12}y^{18}\), is neater and maintains its original value.
Algebraic Expressions Made Simple
Algebraic expressions, like \((-3 x^{4} y^{6})^{3}\), contain constants, variables, and exponents.

Understanding these components helps simplify complex problems.
  • Constants: Numbers with a fixed value, such as -3 in our expression.
  • Variables: Symbols like \(x\) or \(y\) that can represent different values.
  • Exponents: These indicate how many times a number or variable is multiplied by itself.
By identifying each, and knowing how they interact (like using the power rule), you can easily transform and simplify expressions.

This foundational understanding empowers you to handle more complex algebra with confidence.

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Most popular questions from this chapter

Describe what it means to rationalize a denominator. Use both \(\frac{1}{\sqrt{5}}\) and \(\frac{1}{5+\sqrt{5}}\) in your explanation.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. First factoring out the greatest common factor makes it easier for me to determine how to factor the remaining factor, assuming that it is not prime.

Putting Numbers into Perspective. A large number can be put into perspective by comparing it with another number. For example, we put the \(\$ 15.2\) trillion national debt (Example 12 ) and the \(\$ 2.17\) trillion the government collected in taxes (Exercise 115 ) by comparing these numbers to the number of U.S. citizens. For this project, each group member should consult an almanac, a newspaper, or the Internet to find a number greater than one million. Explain to other members of the group the context in which the large number is used. Express the number in scientific notation. Then put the number into perspective by comparing it with another number.

The early Greeks believed that the most pleasing of all rectangles were golden rectangles, whose ratio of width to height is $$\frac{w}{h}=\frac{2}{\sqrt{5}-1}$$ The Parthenon at Athens fits into a golden rectangle once the triangular pediment is reconstructed. (IMAGE CANT COPY) Rationalize the denominator of the golden ratio. Then use a calculator and find the ratio of width to height, correct to the nearest hundredth, in golden rectangles.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}=49$$
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