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Chapter 2: Problem 5

Evaluate the given expression. Do not use a calculator. $$ \left(\frac{2}{3}\right)^{-4} $$

Short Answer

Expert verified
The given expression can be rewritten as \(\frac{1}{\left(\frac{2}{3}\right)^{4}}\). Simplifying further, it becomes \(\frac{1}{\frac{16}{81}}\), which is equal to \(\frac{81}{16}\).

Step by step solution

01

Apply the negative exponent rule

We will use the negative exponent rule to convert the expression into a form that is easier to work with. The rule states that \(a^{-n} = \frac{1}{a^n}\). Applying this to our expression, we get: \[ \left(\frac{2}{3}\right)^{-4} = \frac{1}{\left(\frac{2}{3}\right)^{4}} \]
02

Simplify the expression

Now, we need to simplify the expression by raising the fraction to the power of 4: \[ \frac{1}{\left(\frac{2}{3}\right)^{4}} = \frac{1}{\left(\frac{2^4}{3^4}\right)} \]
03

Calculate the powers

Next, we will calculate \(2^4\) and \(3^4\) to obtain: \[ \frac{1}{\left(\frac{2^4}{3^4}\right)} = \frac{1}{\left(\frac{16}{81}\right)} \]
04

Invert and multiply the fractions

To complete the evaluation, we will invert the fraction in the denominator and multiply: \[ \frac{1}{\frac{16}{81}} = \frac{1}{1} \times \frac{81}{16} = \frac{81}{16} \] The expression \(\left(\frac{2}{3}\right)^{-4}\) evaluates to \(\frac{81}{16}\).

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Most popular questions from this chapter

Suppose \(p\) and \(q\) are polynomials of degree 3 such that \(p(1)=q(1), p(2)=q(2), p(3)=q(3),\) and \(p(4)=q(4) .\) Explain why \(p=q\).

Suppose \(p(x)=2 x^{4}+9 x^{3}+1\) (a) Show that if \(\frac{M}{N}\) is a zero of \(p\), then $$ 2 M^{4}+9 M^{3} N+N^{4}=0 $$ (b) Show that if \(M\) and \(N\) are integers with no common factors and \(\frac{M}{N}\) is a zero of \(p\), then \(M=-1\) or \(M=1\). (c) Show that if \(M\) and \(N\) are integers with no common factors and \(\frac{M}{N}\) is a zero of \(p\), then \(N=-2\) or \(N=2\) or \(N=-1\) or \(N=1\). (d) Show that \(-\frac{1}{2}\) is the only rational zero of \(p\).

Suppose \(t(x)=\frac{5}{4 x^{3}+3}\). (a) Show that the point (-1,-5) is on the graph of \(t\) (b) Give an estimate for the slope of a line containing (-1,-5) and a point on the graph of \(t\) very close to (-1,-5)

A textbook states that the rabbit population on a small island is observed to be $$ 1000+120 t-0.4 t^{4} $$ where \(t\) is the time in months since observations of the island began. Explain why the formula above cannot correctly give the number of rabbits on the island for large values of \(t\).

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (t \circ r)(x) $$
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