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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
\( \left( \frac{16}{25} \right)^{3/2} = \frac{64}{125} \).

Step by step solution

01

Understand the Expression

The expression given is \( \left( \frac{16}{25} \right)^{3/2} \). This means we have to evaluate the fraction \( \frac{16}{25} \) raised to the 3/2 power.
02

Apply the Fractional Exponent Rule

Fractional exponents can be rewritten as roots. Specifically, \( x^{m/n} = (\sqrt[n]{x})^m \). Thus, \( \left( \frac{16}{25} \right)^{3/2} \) becomes \( \left( \sqrt{\frac{16}{25}} \right)^3 \).
03

Simplify the Square Root

Calculate the square root of the fraction: \( \sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} \). This equals \( \frac{4}{5} \), since \( \sqrt{16} = 4 \) and \( \sqrt{25} = 5 \).
04

Raise to the Power of 3

Now, raise the result from the previous step to the power of 3: \( \left( \frac{4}{5} \right)^3 = \frac{4^3}{5^3} = \frac{64}{125} \).
05

Final Evaluation

The simplified result of the expression \( \left( \frac{16}{25} \right)^{3/2} \) is \( \frac{64}{125} \).

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Key Concepts

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

Exponentiation
Exponentiation is a fundamental mathematical operation that involves raising a number, known as the base, to the power of an exponent. This operation is denoted as \( a^b \), where \( a \) is the base and \( b \) is the exponent.
  • If \( b \) is a positive integer, the operation implies multiplying the base \( a \), \( b \) times.
  • If \( b \) is a fraction, it indicates a combination of rooting and raising to a power.
For example, in the expression \( \left( \frac{16}{25} \right)^{3/2} \), the base \( \frac{16}{25} \) is raised to the power of \( \frac{3}{2} \). This separates into two operations: initially finding the root and then exponentiation. Understanding these steps is crucial for evaluating fractional exponents.
Radicals and Roots
When dealing with fractional exponents, knowing how to work with radicals and roots is essential. A radical expression typically uses the square root symbol \( \sqrt{} \), but can also involve cube roots \( \sqrt[3]{} \), fourth roots, and so on.
  • A square root is essentially finding a number that, when multiplied by itself, gives back the original number. For instance, \( \sqrt{16} = 4 \) because \( 4 \times 4 = 16 \).
  • When applying the radical to a fraction like \( \frac{16}{25} \), we treat the numerator and denominator separately: \( \sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5} \).
Understanding radicals makes it easier to simplify expressions that involve roots, particularly when they appear as fractional exponents.
Simplifying Fractions
Simplifying fractions is an important skill when evaluating mathematical expressions. A simplified fraction is one where the numerator and denominator have no common factors other than 1.
  • To simplify, divide both the numerator and the denominator by their greatest common factor (GCF).
  • In terms of exponential expressions, raising a fraction to a power involves exponentiating both the numerator and the denominator independently.
In the expression \( \left( \frac{4}{5} \right)^3 \), it simplifies by computing \( \frac{4^3}{5^3} = \frac{64}{125} \). Neither 64 nor 125 have any common factors, so the fraction is already simplified.
Simplifying properly ensures clarity and accuracy in problem-solving, particularly when combined with other operations such as exponentiation and rooting.

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Most popular questions from this chapter

Identify each function as a polynomial, a rational function, an exponential function, a piecewise linear function, or none of these. (Do not graph them; just identify their types.) $$ f(x)=x^{2}+9 $$

For the functions \(y_{1}=\left(\frac{1}{3}\right)^{x}, y_{2}=\left(\frac{1}{2}\right)^{x}\), \(y_{3}=2^{x}\), and \(y_{4}=3^{x}\) a. Predict which curve will be the highest for large values of \(x\). b. Predict which curve will be the lowest for large values of \(x\). c. Check your predictions by graphing the functions on the window \([-3,3]\) by \([0,5]\). d. From your graph, what is the common \(y\) -intercept? Why do all such exponential functions meet at this point?

For each pair of functions \(f(x)\) and \(g(x)\), find a. \(f(g(x))\) b. \(g(f(x))\) and c. \(f(f(x))\) $$ f(x)=\sqrt{x} ; g(x)=x^{3}-1 $$

Identify each function as a polynomial, a rational function, an exponential function, a piecewise linear function, or none of these. (Do not graph them; just identify their types.) $$ f(x)=x+2 $$

ECONOMICS: Income Tax The following function expresses an income tax that is \(15 \%\) for incomes below $$\$ 6000$$, and otherwise is $$\$ 900$$ plus \(40 \%\) of income in excess of $$\$ 6000$$. \(f(x)=\left\\{\begin{array}{ll}0.15 x & \text { if } 0 \leq x<6000 \\\ 900+0.40(x-6000) & \text { if } x \geq 6000\end{array}\right.\) a. Calculate the tax on an income of $$\$ 3000$$. b. Calculate the tax on an income of $$\$ 6000$$. c. Calculate the tax on an income of $$\$ 10,000$$. d. Graph the function.
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