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Chapter 1: Problem 46

Evaluate each expression without using a calculator. $$ \left(\frac{16}{9}\right)^{-3 / 2} $$

Short Answer

Expert verified
\( \left(\frac{16}{9}\right)^{-\frac{3}{2}} = \frac{27}{64} \).

Step by step solution

01

Understand the Expression

The given expression is \( \left(\frac{16}{9}\right)^{-\frac{3}{2}} \). This is a power expression which involves both a fraction and a negative exponent. Our task is to simplify it without using a calculator.
02

Address the Negative Exponent

A negative exponent implies taking the reciprocal of the base. Hence, \( \left(\frac{16}{9}\right)^{-\frac{3}{2}} = \left(\frac{9}{16}\right)^{\frac{3}{2}} \).
03

Simplify the Fraction

We now have \( \left(\frac{9}{16}\right)^{\frac{3}{2}} \). This can be broken down as \( \left(\frac{9}{16}\right)^{\frac{1}{2}} \) and then cubed. First, find the square root of the fraction: \( \frac{9}{16} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4} \).
04

Cube the Result

Now, take the result from the previous step, \( \frac{3}{4} \), and raise it to the power of 3: \( \left(\frac{3}{4}\right)^3 = \frac{3^3}{4^3} = \frac{27}{64} \).
05

State the Final Result

The simplified form of \( \left(\frac{16}{9}\right)^{-\frac{3}{2}} \) is \( \frac{27}{64} \).

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Key Concepts

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

Negative exponents
Negative exponents can initially seem a bit confusing, but they're easier to understand once you think of them as instructions for inverses. When you see a negative exponent, it tells you to take the reciprocal of the base. For example, if you have an expression like \( a^{-n} \), you simply take the reciprocal of \( a \) and then raise it to the positive power \( n \).
  • Step example: \( \left(\frac{16}{9}\right)^{-\frac{3}{2}} = \left(\frac{9}{16}\right)^{\frac{3}{2}} \)
  • The base \( \frac{16}{9} \) was inversed to \( \frac{9}{16} \).
This is particularly helpful when simplifying expressions, as it allows you to switch the position of numbers in a fraction, turning division into multiplication. Negative exponents thus streamline calculations and can make mathematical expressions easier to work with once properly understood.
Fractional exponents
Fractional exponents are a way to represent roots using the language of powers. A fractional exponent like \( a^{\frac{m}{n}} \) is a convenient notation that combines the concepts of roots and powers together.
  • In the expression \( a^{\frac{m}{n}} \), \( n \) describes the root, and \( m \) describes the power to which the base \( a \) is raised.
  • For \( \left(\frac{9}{16}\right)^{\frac{3}{2}} \), you first take the square root (because of the \( \frac{1}{2} \)), resulting in \( \frac{3}{4} \).
  • Then, raise the result \( \frac{3}{4} \) to the power of 3.
Using fractional exponents can often make calculations more straightforward, especially when dealing with roots and powers in the same expression, without switching between different types of notation.
Simplifying fractions
Simplifying fractions is a fundamental skill in algebra and involves rewriting a fraction to its simplest form. It sounds complex, but involves straightforward steps:
  • First, try to factorize the numerator and the denominator to their prime components if needed.
  • Divide the highest common factor out of both the numerator and the denominator.
For fractional exponent expressions, sometimes simplifying involves dealing with roots. For instance,
  • In \( \left(\frac{9}{16}\right)^{\frac{1}{2}} = \frac{3}{4} \), the square root of each part (\( \sqrt{9} = 3 \) and \( \sqrt{16} = 4 \)) yields the simplified fraction.
Once the fraction is in its simplest form, performing further operations like exponentiation becomes more manageable. This can often involve repeating the process of simplification until the expression cannot be simplified further.

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Most popular questions from this chapter

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ f(x)=7 x^{2}-3 x+2 $$

The following function expresses dog-years as 15 dog-years per human-year for the first year, 9 dog-years per human-year for the second year, and then 4 dog-years per human-year for each year thereafter. \(f(x)=\left\\{\begin{array}{ll}15 x & \text { if } 0 \leq x \leq 1 \\\ 15+9(x-1) & \text { if } 12\end{array}\right.\)

Use your graphing calculator to graph the following four equations simultaneously on the window \([-10,10]\) by \([-10,10]:\) $$ \begin{array}{l} y_{1}=3 x+4 \\ y_{2}=1 x+4 \end{array} $$ \(y_{3}=-1 x+4\) (Use \((-)\) to get \(-1 x\). \()\) $$ y_{4}=-3 x+4 $$ a. What do the lines have in common and how do they differ? b. Write the equation of a line through this \(y\) -intercept with slope \(\frac{1}{2}\). Then check your answer by graphing it with the others.

SOCIAL SCIENCE: Age at First Marriage Americans are marrying later and later. Based on data for the years 2000 to 2007 , the median age at first marriage for men is \(y_{1}=0.12 x+26.8\), and for women it is \(y_{2}=0.12 x+25\), where \(x\) is the number of years since 2000 . a. Graph these lines on the window \([0,30]\) by \([0,35] .\) b. Use these lines to predict the median marriage ages for men and women in the year 2020 . [Hint: Which \(x\) -value corresponds to 2020 ? Then use TRACE, EVALUATE, or TABLE.] c. Predict the median marriage ages for men and women in the year 2030 .

Which of the following is not a polynomial, and why? $$ x^{2}+\sqrt{2} \quad x^{\sqrt{2}}+1 \quad \sqrt{2} x^{2}+1 $$
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