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Chapter 5: Problem 29

For Problems \(1-30\), evaluate each numerical expression. $$ 125^{\frac{4}{3}} $$

Short Answer

Expert verified
125^{4/3}=625

Step by step solution

01

Understand the Expression

The expression given is \( 125^{\frac{4}{3}} \). This notation means taking the number 125 to the power of \( \frac{4}{3} \). It involves both a root and an exponent.
02

Rewrite as a Root and Exponent

The expression \( a^{\frac{m}{n}} \) can be broken down into \( (a^m)^{\frac{1}{n}} \) or \( (a^{\frac{1}{n}})^m \). Here, we can rewrite \( 125^{\frac{4}{3}} \) as \( (125^{\frac{1}{3}})^4 \).
03

Calculate the Cube Root

Find \( 125^{\frac{1}{3}} \). This is the cube root of 125. Since \( 5^3 = 125 \), \( 125^{\frac{1}{3}} = 5 \).
04

Raise to the Power of 4

Now, raise the result from Step 3 to the power of 4. Calculate \( 5^4 \). Since \( 5^4 = 5 \times 5 \times 5 \times 5 = 625 \), the result of \( (125^{\frac{1}{3}})^4 \) is 625.

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Key Concepts

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

Cube Roots
A cube root of a number is a value that, when multiplied by itself three times, gives the original number. Take, for example, the cube root of 125. We look for a number that satisfies this equation:
  • If we multiply the number by itself three times, we obtain 125.
In this case, 5 is the cube root of 125 because \[ 5 imes 5 imes 5 = 125 \]. To express this using radicals, we write it as \( 125^{\frac{1}{3}} \), which signifies the cube root of 125. Understanding cube roots is crucial because they simplify the complex process of breaking down an expression into more manageable terms before further exponentiation.
Simplifying Exponents
Simplifying exponents means breaking down expressions to their most basic form. When working with expressions like \( 125^{\frac{4}{3}} \), distinctly recognize the part that can be simplified. One method is to deal with the exponent by thinking of it as \( \left( 125^{\frac{1}{3}} \right)^4 \). Simplifying here involves evaluating \( 125^{\frac{1}{3}} \) first to obtain \( 5 \) (since \( 5^3 = 125 \)). Raising \( 5 \) to the power of \( 4 \), we find \( 5^4 = 625 \). Thus, simplifying the exponent means methodically breaking the expression into steps to reach the simplest form.
Exponentiation
Exponentiation is the process of raising a base number to a power. This operation compacts repeated multiplication of the same number. In \( 125^{\frac{4}{3}} \), the concept applies in two steps. First, identify that exponentiation covers taking cube roots and raising to powers. To simplify:
  • Identify the base and fractional exponent.
  • Calculate any roots required by the fractional exponent.
  • Raise the simplified result to the power indicated by the exponent鈥檚 numerator.
It turns the potentially complex problem into simpler steps, helping avoid calculation mistakes and improve understanding of the expression's mechanics.
Fractional Exponents
Fractional exponents are a powerful expression tool that blends roots and powers. When you see fractional exponents, think of them as a combination of two operations: finding roots and then exponentiating. The expression \( 125^{\frac{4}{3}} \) is using \( \frac{4}{3} \) as an exponent. This tells us to do:
  • Find the cube root (denoted by \( \frac{1}{3} \)).
  • Raise the result to the fourth power (numerator \( 4 \)).
This method divides the exponent into manageable steps, leading to \( (5)^4 = 625 \). Such simplified breakdown through fractional exponents aids in understanding complex mathematical expressions more clearly.

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Most popular questions from this chapter

For Problems \(1-18\), write each of the following in scientific notation. \(0.0214\)

Use your calculator to estimate each of the following. Express final answers in scientific notation with the number between 1 and 10 rounded to the nearest onethousandth. (a) \(4576^{4}\) (b) \(719^{10}\) (c) \(28^{12}\) (d) \(8619^{6}\) (e) \(314^{5}\) (f) \(145,723^{2}\)

Convert each number to scientific notation and perform the indicated operations. Express the result in ordinary decimal notation. $$ (0.00007)(11,000) $$

Perform the indicated operations and express answers in simplest radical form. $$ \sqrt[3]{5} \sqrt{5} $$

Convert each number to scientific notation and perform the indicated operations. Express the result in ordinary decimal notation. $$ \frac{0.00072}{0.0000024} $$
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