/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Evaluate the given expression. D... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 6

Evaluate the given expression. Do not use a calculator. $$ \left(\frac{5}{4}\right)^{-3} $$

Short Answer

Expert verified
The evaluated expression of \(\left(\frac{5}{4}\right)^{-3}\) without using a calculator is \(\frac{64}{125}\).

Step by step solution

01

Understand the properties of negative exponents

A negative exponent is used to represent the reciprocal of that number raised to the given exponent. It can be seen as a way to reverse the operation of exponentiation. For example, \(a^{-n}\) is the same as \(\frac{1}{a^n}\).
02

Apply the negative exponent to the given expression

In the given expression \(\left(\frac{5}{4}\right)^{-3}\), the base is \(\frac{5}{4}\) and the exponent is -3. We can rewrite the expression using the property of negative exponents. $$ \left(\frac{5}{4}\right)^{-3} = \frac{1}{\left(\frac{5}{4}\right)^3} $$
03

Raise the base to the power of 3

To raise a fraction, such as \(\frac{5}{4}\), to the power of 3, we raise both the numerator and denominator to the power of 3. $$ \left(\frac{5}{4}\right)^3 = \frac{5^3}{4^3} $$
04

Evaluate the powers

Compute the power for the numerator and denominator. $$ 5^3 = 5 \cdot 5 \cdot 5 = 125 \\ 4^3 = 4 \cdot 4 \cdot 4 = 64 $$
05

Write the final result

Combine the results from the previous steps and write down the final expression. $$ \frac{1}{\left(\frac{5}{4}\right)^3} = \frac{1}{\frac{125}{64}} $$ To determine the final value of the expression, simply flip the fraction in the denominator and multiply it by the numerator. This gives: $$ \frac{1}{\frac{125}{64}} = \frac{1}{1} \times \frac{64}{125} = \frac{64}{125} $$ Therefore, the evaluation of the expression \(\left(\frac{5}{4}\right)^{-3}\) without using a calculator is \(\frac{64}{125}\).

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Most popular questions from this chapter

Find all real numbers \(x\) such that $$ x^{4}+5 x^{2}-14=0 $$.

Suppose \(p\) is a polynomial and \(t\) is a number. Explain why there is a polynomial \(G\) such that $$ \frac{p(x)-p(t)}{x-t}=G(x) $$ for every number \(x \neq t\).

Suppose \(s(x)=4 x^{3}-2\) (a) Show that the point (1,2) is on the graph of \(s\). (b) Give an estimate for the slope of a line containing (1,2) and a point on the graph of \(s\) very close to (1,2) [Hint: Use the result of Exercise \(18 .]\)

Find a number \(b\) such that 3 is a zero of the polynomial \(p\) defined by $$ p(x)=1-4 x+b x^{2}+2 x^{3} $$.

Suppose \(M\) and \(N\) are odd integers. Explain why $$ x^{2}+M x+N $$ has no rational zeros.
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