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Chapter 8: Problem 34

Write as a radical and then simplify. \((8125) 1 / 3\)

Short Answer

Expert verified
The expression simplifies to \(5 \sqrt[3]{65}\).

Step by step solution

01

Write as a Radical

The expression \(8125^{1/3}\) can be rewritten using radical notation. In this case, \(8125^{1/3}\) is equivalent to the cube root of 8125, which can be written as \(\sqrt[3]{8125}\).
02

Simplify the Radical Expression

To simplify \(\sqrt[3]{8125}\), we first need to find if 8125 has any perfect cube factors. \(8125 = 5^4 imes 13\). Here, \(125 = 5^3\) is a perfect cube. Therefore, rewriting the expression using this factorization gives us \(\sqrt[3]{125 imes 65}\).
03

Evaluate the Cube Root of Perfect Cubes

The cube root of \(125\) is \(5\) because \(5^3 = 125\). The remaining factor outside of the cube is \(65\), which can be written as \(\sqrt[3]{65}\).
04

Combine Results

The simplified form of the radical expression is thus \(5 \cdot \sqrt[3]{65}\). Therefore, the simplified cube root of 8125 is \(5 \sqrt[3]{65}\).

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Key Concepts

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

Cube Roots
The cube root of a number is the value that, when multiplied by itself three times, gives the original number. It's represented by the symbol \(\sqrt[3]{}\). For example, in \(\sqrt[3]{8125}\), we are finding the number that if cubed equals 8125.
  • To simplify, look for perfect cubes, like numbers raised to the third power, within the original number.
  • Finding a perfect cube factor can greatly simplify the expression.
In our case, \(125\) is a perfect cube because \(5^3 = 125\). So, \(\sqrt[3]{125}\) simplifies to 5. The remaining factor doesn't form a perfect cube and is left in radical form as \(\sqrt[3]{65}\).
Exponents
Exponents are a way to express repeated multiplication. An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in \(8125^{1/3}\), 8125 is the base, and \(1/3\) is the exponent.
  • The exponent \(1/3\) signifies the cube root, converting it into radical notation.
  • Exponents can help simplify complex calculations using rules like \((a^m)^n = a^{m*n}\).
Converting \(8125^{1/3}\) into \(\sqrt[3]{8125}\) lets us factor and simplify it further by finding cube roots.
Factorization
Factorization involves breaking down a number into its prime factors, which can be particularly useful when simplifying expressions involving roots. To factorize 8125, we need to express it in terms of its prime factors
  • First, identify if there are any repeated factors that can form a perfect cube. For \(8125\), this factorization leads us to \(5^4 \times 13\).
  • More precisely, \(125 = 5^3\) is a perfect cube, helping simplify the radical expression.
By recognizing \(5^3\) in 8125, we can simplify the cube root to \(5 \cdot \sqrt[3]{65}\). This step reduces the complexity of the expression by converting a portion into a whole number.

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