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Chapter 0: Problem 37

Use the laws of exponents to compute the numbers. $$\left(\frac{8}{27}\right)^{2 / 3}$$

Short Answer

Expert verified
\( \frac{4}{9} \)

Step by step solution

01

- Understand the Exponent Fraction

The given expression is \(\left(\frac{8}{27}\right)^{2 / 3}\). Here, the exponent \( \frac{2}{3} \) indicates that we need to first take the cube root (since the denominator is 3) and then square the result (since the numerator is 2).
02

- Compute the Cube Root

First, compute the cube root of both the numerator and the denominator. The cube root of 8 is 2 (because \(2^3 = 8\)) and the cube root of 27 is 3 (because \(3^3 = 27\)). So, \(\left(\frac{8}{27}\right)^{1/3} = \frac{2}{3}\).
03

- Square the Result

Next, square the result obtained in Step 2. Therefore, \( \left(\frac{2}{3}\right)^2 \) is \((2^2)/(3^2) = \frac{4}{9}\).

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Key Concepts

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

Fractional Exponents
Fractional exponents may seem complex, but they are quite manageable once you break them down. The fraction in the exponent tells you two things:
  • The numerator (top number) is the power you will raise the number to.
  • The denominator (bottom number) tells you which root to take.
In the given exercise, \(\frac{2}{3}\) is the fractional exponent. This means:
  • First, you take the cube root (since the denominator is 3).
  • Then, you square the result (since the numerator is 2).
Using these steps helps to simplify expressions that seem complex at first glance.
Cube Root
Finding a cube root means you are looking for a number that, when multiplied by itself three times, gives the original number.
In our exercise, we need to find the cube root of both the numerator and the denominator of the fraction \(\frac{8}{27}\).
  • The cube root of 8 is 2 because \[2^3 = 8\]
  • The cube root of 27 is 3 because \[3^3 = 27\]
After finding these cube roots, you obtain a simpler fraction: \(\frac{2}{3}\). Knowing how to work with cube roots helps in simplifying more complex fractional exponents.
Squaring Fractions
Squaring a fraction involves raising both the numerator and the denominator to the power of 2.
In our example, we got \(\frac{2}{3}\) from the cube root step. Now, we need to square it:
  • Square the numerator: \[2^2 = 4\]
  • Square the denominator: \[3^2 = 9\]
So, \[ \left(\frac{2}{3}\right)^2 = \frac{4}{9}\]. By understanding how to square a fraction, you can easily handle expressions that combine roots and exponents, making seemingly complex operations straightforward.

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Most popular questions from this chapter

Let \(f(x)=x^{2} .\) Graph the functions \(f(x+1)\) \(f(x-1), f(x+2),\) and \(f(x-2) .\) Make a guess about the relationship between the graph of a general function \(f(x)\) and the graph of \(f(g(x)),\) where \(g(x)=x+a\) for some constant \(a\) Test your guess on the functions \(f(x)=x^{3}\) and \(f(x)=\sqrt{x}.\)

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\sqrt{1+x}(1+x)^{3 / 2}$$

Use your graphing calculator to find the value of the given function at the indicated values of \(x .\) $$f(x)=x^{4}+2 x^{3}+x-5 ; \quad x=-\frac{1}{2}, x=3$$

Let \(f(x)=\sqrt[3]{x}\) and \(g(x)=\frac{1}{x^{2}} .\) Calculate the following functions. Take \(x > 0\). $$\frac{f(x)}{g(x)}$$

Calculate the compound amount from the given data. principal \(=\$ 20,000,\) compounded quarterly, 3 years, annual rate \(=12 \%\)
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