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Chapter 1: Problem 17

Evaluate each expression without using a calculator. $$ 25^{1 / 2} $$

Short Answer

Expert verified
\( 25^{1/2} = 5 \).

Step by step solution

01

Understanding the Expression

The expression given is \( 25^{1/2} \). The notation \( a^{1/2} \) refers to the square root of \( a \). So, \( 25^{1/2} \) means the square root of 25.
02

Identify the Square Root

We need to find the number which, when multiplied by itself, equals 25. The number that satisfies this condition is 5, since \( 5 \times 5 = 25 \).
03

Finalize the Answer

Since the square root of 25 is 5, we conclude that \( 25^{1/2} = 5 \).

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Key Concepts

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

Exponents
Exponents are a fundamental concept in mathematics that represent repeated multiplication. When we encounter a number expressed as a base with an exponent, like \( a^n \), it means we multiply the base \( a \) by itself \( n \) times. For example, \( 3^4 \) means \( 3 \times 3 \times 3 \times 3 \). There are a few special cases to be aware of:
  • Any number to the power of 1 is the number itself (e.g., \( 7^1 = 7 \)).
  • Any number to the power of 0 is 1, except for 0 itself (e.g., \( 9^0 = 1 \)).
A fractional exponent, like \( a^{1/n} \), signifies the \( n \)-th root of \( a \). Therefore, \( 25^{1/2} \) is the square root of 25. Understanding these basics is crucial when working with expressions involving exponents.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form so they are easy to work with or interpret. For expressions involving exponents, simplification can include applying rules of exponents and roots.
  • When faced with a fractional exponent like \( x^{1/2} \), it indicates the square root, meaning you find the number that produces \( x \) when squared.
  • For example, in \( 25^{1/2} \), simplify by identifying that 5 is the square root of 25, as \( 5 \times 5 = 25 \).
  • Always check if you can further reduce or simplify. Sometimes expressions might already be in their simplest form.
By practicing these steps, you can become more efficient at working with expressions, making complex problems manageable.
Mathematical Notation
Mathematical notation is a language of symbols used to express concepts in mathematics succinctly and accurately. Clear understanding of this notation helps solve and communicate mathematical problems.
  • For example, the expression \( 25^{1/2} \) uses the notation for exponents to indicate the square root of 25.
  • Common symbols include:
    • \( +, -, \times, \div \) for basic operations.
    • \( ^ \) to denote exponents.
    • \( \sqrt{} \) to signify roots.
  • Understanding these symbols allows you to interpret and solve problems systematically.
Mathematical notation streamlines the process of dealing with complex operations, and learning it is like learning a universal mathematical language.

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Most popular questions from this chapter

BUSINESS: Cost Functions A company manufactures bicycles at a cost of \(\$ 55\) each. If the company's fixed costs are \(\$ 900\), express the company's costs as a linear function of \(x\), the number of bicycles produced.

BIOMEDICAL: Cell Growth One leukemic cell in an otherwise healthy mouse will divide into two cells every 12 hours, so that after \(x\) days the number of leukemic cells will be \(f(x)=4^{x}\). a. Find the approximate number of leukemic cells after 10 days. b. If the mouse will die when its body has a billion leukemic cells, will it survive beyond day \(15 ?\)

BUSINESS: MBA Salaries Starting salaries in the United States for new recipients of MBA (master of business administration) degrees have been rising approximately linearly, from \(\$ 78,040\) in 2005 to \(\$ 89,200\) in \(2010 .\) a. Use the two (year, salary) data points \((0,78.0)\) and \((5,89.2)\) to find the linear relationship \(y=m x+b\) between \(x=\) years since 2005 and \(y=\) salary in thousands of dollars. b. Use your formula to predict a new MBA's salary in 2020 . [Hint: Since \(x\) is years after 2005, what \(x\) -value corresponds to \(2020 ?]\)

BUSINESS: Research Expenditures An electronics company's research budget is \(R(p)=3 p^{0.25}\), where \(p\) is the company's profit, and the profit is predicted to be \(p(t)=55+4 t\), where \(t\) is the number of years from now. (Both \(R\) and \(p\) are in millions of dollars.) Express the research expenditure \(R\) as a function of \(t\), and evaluate the function at \(t=5\).

For any \(x\), the function \(\operatorname{INT}(x)\) is defined as the greatest integer less than or equal to \(x\). For example, \(\operatorname{INT}(3.7)=3\) and \(\operatorname{INT}(-4.2)=-5\) a. Use a graphing calculator to graph the function \(y_{1}=\operatorname{INT}(x) .\) (You may need to graph it in DOT mode to eliminate false connecting lines.)
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