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Chapter 8: Problem 67

Complete the statement using \(>\) or \(<\). $$ \left(7^{2}\right)^{3} \geq 7^{5} $$

Short Answer

Expert verified
Based on the solution steps, \((7^{2})^{3}\) is greater than \(7^{5}\), so the completed statement is \((7^{2})^{3} > 7^{5}\).

Step by step solution

01

Simplify the first expression

Begin by simplifying the first expression. An exponential expression can be simplified by multiplying the exponents, so \((7^{2})^{3}\) simplifies to \(7^{2*3}\) which is equal to \(7^{6}\).
02

Compare the expressions

Now that we have simplified the first expression to \(7^{6}\), we can compare this to the second expression, \(7^{5}\). Since \(7^{6}\) as the base 7 is being raised to a higher power than \(7^{5}\), it is clear that \(7^{6} > 7^{5}\).

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Key Concepts

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

Simplifying Expressions
When dealing with exponents, one of the basic skills is to simplify expressions. Simplification makes complex problems easier to understand and solve. By mastering this, students can tackle more challenging mathematical tasks with ease. Here's a way to think about it:
  • Exponents are a shorthand for repeated multiplication. For example, \(7^3\) means you multiply 7 by itself three times, or \(7 \times 7 \times 7\).
  • To simplify an expression like \((7^2)^3\), we apply the rule where you multiply the exponents. So, \((7^2)^3\) turns into \(7^{2 \times 3}\), and that equals \(7^6\).
  • This rule \((a^m)^n = a^{m \times n}\) helps transform seemingly complex expressions into something more manageable.
With these steps, expressions involving powers can be broken down and greatly simplified.
Inequalities
Inequalities are mathematical statements that express the relationship of one quantity being greater than, less than, or equal to another. These comparisons are essential in understanding ranges and limits in math.
  • The symbols used in inequalities are \(>\) for "greater than" and \(<\) for "less than."
  • In an exercise where we need to complete a statement like \((7^6) \geq (7^5)\), we evaluate whether the given condition (greater than or equal to) holds true after simplification.
  • Understanding which of two numbers is larger is at the core of solving inequalities and greatly influences decision-making in calculations.
By practicing with inequalities, students become adept at evaluating and comparing sizes of numbers and expressions.
Comparing Powers
Comparing powers involves assessing expressions to see which has a larger or smaller magnitude. This skill is useful not just in academic mathematics but also in practical applications like computing and science.
  • Expressions with smaller bases raised to higher exponents can surpass those with larger bases raised to smaller exponents. This is the essence of comparing powers.
  • In our exercise, we compared \(7^6\) with \(7^5\). Since both have the same base, the exponent value directly determines which is larger. Hence, \(7^6 > 7^5\).
  • When based and exponent values are different, comparing powers might require additional steps like finding common bases or further simplification.
Mastering the technique of comparing powers prepares students to handle sophisticated mathematical challenges effectively.

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Most popular questions from this chapter

Use the following information. The power generated by a windmill can be modeled by the equation \(w=0.015 s^{3},\) where \(w\) is the power measured in watts and \(s\) is the wind speed in miles per hour. Part A of a test has 10 true-false questions. Part B has 10 multiple-choice questions. Each of the multiple-choice questions has 4 possible answers. There are \(2^{10}\) ways to answer the 10 questions in Part A. There are \(4^{10}\) ways to answer the 10 questions in Part B. a. How many ways are there to answer all 20 questions? b. If you guess the answer to each question, what is the probability that you will get them all right?

Simplify the expression. Then use a calculator to evaluate the expression. Round the result to the nearest tenth when appropriate. $$ 6.5^{3} \cdot 6.5^{4} $$

Draw a box-and-whisker plot of the data. \(48,10,48,25,40,42,44,23,21,13,50,17\)

Evaluate the expression. \(\frac{24}{x^{3}}\) when \(x=2\)

Sketch the graph of the inequality. $$x>15$$
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