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Chapter 1: Problem 28

Evaluate each expression without using a calculator. $$ \left(\frac{16}{25}\right)^{3 / 2} $$

Short Answer

Expert verified
\( \frac{64}{125} \)

Step by step solution

01

Understand the Expression

The expression given is \( \left(\frac{16}{25}\right)^{3/2} \). This means we are dealing with a fractional base raised to a fractional exponent, which can be broken down into more straightforward components.
02

Simplify the Fraction

Notice that \( \frac{16}{25} \) can be written as \( \left(\frac{4}{5}\right)^2 \) because \( 16 = 4^2 \) and \( 25 = 5^2 \). So the expression becomes \( \left(\left(\frac{4}{5}\right)^2\right)^{3/2} \).
03

Apply Exponent Combination Rule

Use the rule \( (a^m)^n = a^{m \cdot n} \) to simplify the expression. Here, \( m = 2 \) and \( n = \frac{3}{2} \), so we have \( \left(\frac{4}{5}\right)^{2 \times \frac{3}{2}} = \left(\frac{4}{5}\right)^3 \).
04

Evaluate the Exponent

Now, compute \( \left(\frac{4}{5}\right)^3 \). This is calculated by raising both the numerator and the denominator to the power of 3: \( 4^3 = 64 \) and \( 5^3 = 125 \). Thus, \( \left(\frac{4}{5}\right)^3 = \frac{64}{125} \).
05

Write the Final Answer

The expression \( \left(\frac{16}{25}\right)^{3/2} \) simplifies to \( \frac{64}{125} \).

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Key Concepts

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

Fractional Base
A fractional base refers to an expression where the base is a fraction. In our exercise, the base is \( \frac{16}{25} \). When dealing with fractional bases, the goal is often to simplify them to make exponentiation easier. This involves expressing both the numerator and denominator as powers of smaller numbers if possible. For example, \( 16 \) is \( 4^2 \) and \( 25 \) is \( 5^2 \). Thus, we can rewrite \( \frac{16}{25} \) as \( \left(\frac{4}{5}\right)^2 \). This simplification is useful because it makes applying exponent rules straightforward.
Exponentiation Rules
Whenever an expression involves exponents, certain rules apply. These rules help transform the expression into a manageable form. One such rule is the power of a power rule: \( (a^m)^n = a^{m \cdot n} \). This rule simplifies an expression with a power raised to another power. In our exercise, we applied this rule to \( \left(\left(\frac{4}{5}\right)^2\right)^{3/2} \). Here, multiplying the exponents \( 2 \) and \( \frac{3}{2} \) results in \( 2 \times \frac{3}{2} = 3 \). Therefore, the expression simplifies down to \( \left(\frac{4}{5}\right)^3 \). Using this rule can effectively break down complex exponentiation problems into simpler steps.
Simplifying Fractions
Simplifying fractions involves reducing them to their simplest form. Once you have an expression like \( \left(\frac{4}{5}\right)^3 \), you simplify by calculating the power of both the numerator and the denominator. Raise \( 4 \) to the third power to get \( 64 \), and \( 5 \) to the third power to get \( 125 \). Thus, \( \left(\frac{4}{5}\right)^3 = \frac{64}{125} \). By ensuring each component of the fraction is in its simplest form, you ensure the resulting fraction is as reduced as possible. This helps in presentations and further calculations.

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Most popular questions from this chapter

When defining \(x^{m / n}\), why did we require that the exponent \(\frac{\mathrm{m}}{\mathrm{u}}\) be fully reduced? [Hint: \((-1)^{2 / 3}=(\sqrt[3]{-1})^{2}=1,\) but with an equal but unreduced exponent you get \((-1)^{4 / 6}=(\sqrt[6]{-1})^{4}\). Is this defined?]

We have discussed quadratic functions that open \(u p\) or open down. Can a quadratic function open sideways? Explain.

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 \(t=10 .\)

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.) b. From your graph, what are the domain and range of this function?

91-92. BUSINESS: Phillips Curves Unemployment and inflation are inversely related, with one rising as the other falls, and an equation giving the relation is called a Phillips curve after the economist A. W. Phillips (1914-1975). Phillips used data from 1861 to 1957 to establish that in the United Kingdom the unemployment rate \(x\) and the wage inflation rate \(y\) were related by $$ y=9.638 x^{-1394}-0.900 $$ where \(x\) and \(y\) are both percents. Use this relation to estimate the inflation rate when the unemployment rate was a. 2 percent b. 5 percent
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