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Chapter 1: Problem 34

Evaluate each expression without using a calculator. $$ 9^{-1 / 2} $$

Short Answer

Expert verified
\(\frac{1}{3}\)

Step by step solution

01

Understand the Expression

The expression given is \(9^{-1/2}\). The exponent here is \(-1/2\), which can be interpreted as a negative fractional exponent.
02

Convert the Negative Exponent

Recall that a negative exponent indicates the reciprocal of the base raised to the positive of that exponent. For example, \(a^{-b} = \frac{1}{a^b}\). Applying this to the expression \(9^{-1/2}\) gives \(\frac{1}{9^{1/2}}\).
03

Simplify the Fractional Exponent

A fractional exponent \(a^{1/2}\) represents a square root. Hence, \(9^{1/2}\) is the same as \(\sqrt{9}\).
04

Calculate the Square Root

\(\sqrt{9} = 3\), so we substitute this back into the expression to get \(\frac{1}{3}\).
05

Write the Final Answer

The expression \(9^{-1/2}\) simplifies to \(\frac{1}{3}\).

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Key Concepts

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

Negative Exponents
Let's understand negative exponents. In mathematics, an exponent indicates the number of times a base is multiplied by itself. For example, in \(3^2\), 3 is multiplied by itself twice to give \(9\).
However, when the exponent is negative, like in \(a^{-1}\), this signifies the reciprocal of the base raised to the positive exponent. Simply put, \(a^{-b} = \frac{1}{a^b}\).
  • Instead of multiplication, think about division or taking the reciprocal.
  • This is how we transform \(9^{-1/2}\) into \(\frac{1}{9^{1/2}}\).
Understanding this rule helps simplify expressions without direct calculation, underscoring an essential part of exponent knowledge.
Reciprocal
The concept of the reciprocal is quite straightforward once you break it down. A reciprocal of a number is essentially flipping the numerator and the denominator of a fraction.
For any non-zero number \(a\), the reciprocal is \(\frac{1}{a}\). For instance, the reciprocal of 5 is \(\frac{1}{5}\).
  • When you encounter expressions like \(9^{-1/2}\), the first action is to find the reciprocal.
  • This results in \(\frac{1}{9^{1/2}}\), simplifying our process by setting up for further calculations.
Reciprocals are vital when dealing with divisions and negative exponents. They are the stepping stone that paves the way to efficiently simplifying an expression.
Square Roots
Square roots play a crucial role when dealing with fractional exponents. A square root occurs when you find a number which, when multiplied by itself, gives the original number.
For example, the square root of 9 is 3 because \(3 \times 3 = 9\). In mathematical terms, it is expressed as \(\sqrt{9} = 3\) or \(9^{1/2} = 3\).
  • Fractional exponents, like \(9^{1/2}\), are interpreted as square roots.
  • This understanding makes it clear why \(\sqrt{9}\) is used in simplifying \(9^{-1/2}\).
Once you simplify the square root, you can proceed with the rest of the problem smoothly, as seen in reducing \(9^{1/2}\) to obtain \(\frac{1}{3}\) in the original expression.

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Most popular questions from this chapter

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ f(x)=3 x^{2}-5 x+2 $$

SOCIAL SCIENCE: Equal Pay for Equal Work Women's pay has often lagged behind men's, although Title VII of the Civil Rights Act requires equal pay for equal work. Based on data from \(2000-2008\), women's annual earnings as a percent of men's can be approximated by the formula \(y=0.51 x+77.2, \quad\) where \(x\) is the number of years since 2000 . (For example, \(x=10\) gives \(y=82.3\), so in 2010 women's wages were about \(82.3 \%\) of men's wages.) a. Graph this line on the window \([0,30]\) by \([0,100]\). b. Use this line to predict the percentage in the year 2020\. [Hint: Which \(x\) -value corresponds to 2020 ? Then use TRACE, EVALUATE, or TABLE.] c. Predict the percentage in the year 2025 .

Use the TABLE feature of your graphing calculator to evaluate \(\left(1+\frac{1}{x}\right)^{x}\) for values of \(x\) such as \(100,10,000,1,000,000\), and higher values. Do the resulting numbers seem to be approaching a limiting value? Estimate the limiting value to five decimal places. The number that you have approximated is denoted \(e\), and will be used extensively in Chapter \(4 .\)

An insurance company keeps reserves (money to pay claims) of \(R(v)=2 v^{0.3}\), where \(v\) is the value of all of its policies, and the value of its policies is predicted to be \(v(t)=60+3 t\), where \(t\) is the number of years from now. (Both \(R\) and \(v\) are in millions of dollars.) Express the reserves \(R\) as a function of \(t\). and evaluate the function at \(t=10\).

BUSINESS: Isocost Lines An isocost line (iso means "same") shows the different combinations of labor and capital (the value of factory buildings, machinery, and so on) a company may buy for the same total cost. An isocost line has equation $$ w L+r K=\mathrm{C} \quad \text { for } L \geq 0, \quad K \geq 0 $$ where \(L\) is the units of labor costing \(w\) dollars per unit, \(K\) is the units of capital purchased at \(r\) dollars per unit, and \(C\) is the total cost. Since both \(L\) and \(K\) must be nonnegative, an isocost line is a line segment in just the first quadrant. a. Write the equation of the isocost line with \(w=8, \quad r=6, \quad C=15,000\), and graph it in the first quadrant. b. Verify that the following \((L, K)\) pairs all have the same total cost. \((1875,0),(1200,900),(600,1700),(0,2500)\)
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