Understanding the Cube Root Problem

The question asks us to find the value of the cube root of a mixed number, specifically \(\sqrt[3]{{4\frac{{12}}{{125}}}}\). To solve this, we first need to convert the mixed number into an improper fraction and then find the cube root of the resulting fraction.

Step-by-Step Solution for Finding the Cube Root

Step 1: Convert the mixed number to an improper fraction.

The given mixed number is \(4\frac{{12}}{{125}}\). To convert this to an improper fraction, we multiply the whole number part by the denominator and add the numerator. The denominator remains the same.

Calculation:

\(4\frac{{12}}{{125}} = \frac{(4 \times 125) + 12}{125} = \frac{500 + 12}{125} = \frac{512}{125}\)

So, the expression becomes \(\sqrt[3]{{\frac{512}{125}}}\).

Step 2: Find the cube root of the improper fraction.

The cube root of a fraction is equal to the cube root of the numerator divided by the cube root of the denominator. That is, \(\sqrt[3]{{\frac{a}{b}}} = \frac{{\sqrt[3]{a}}}{{\sqrt[3]{b}}}\).

Applying this rule:

\(\sqrt[3]{{\frac{512}{125}}} = \frac{{\sqrt[3]{512}}}{{\sqrt[3]{125}}}\)

Step 3: Find the cube roots of the numerator and the denominator.

We need to find the numbers that, when multiplied by themselves three times, give 512 and 125.

• For the numerator, 512: We look for a number \(x\) such that \(x \times x \times x = 512\). By trial or knowing cube values, we find \(8 \times 8 \times 8 = 64 \times 8 = 512\). So, \(\sqrt[3]{512} = 8\).
• For the denominator, 125: We look for a number \(y\) such that \(y \times y \times y = 125\). We know \(5 \times 5 \times 5 = 25 \times 5 = 125\). So, \(\sqrt[3]{125} = 5\).

Now substitute these values back into the expression:

\(\frac{{\sqrt[3]{512}}}{{\sqrt[3]{125}}} = \frac{8}{5}\)

Step 4: Convert the resulting improper fraction back to a mixed number (if necessary).

The fraction \(\frac{8}{5}\) is an improper fraction because the numerator (8) is greater than the denominator (5). To convert it to a mixed number, divide the numerator by the denominator.

\(8 \div 5\)

\(8\) divided by \(5\) is \(1\) with a remainder of \(3\).

So, \(\frac{8}{5}\) as a mixed number is \(1\) (the quotient) and \(\frac{3}{5}\) (the remainder over the original denominator).

\(\frac{8}{5} = 1\frac{3}{5}\)

Thus, the value of \(\sqrt[3]{{4\frac{{12}}{{125}}}}\) is \(1\frac{3}{5}\).

Understanding Cube Roots and Fractions

A cube root is the value that, when multiplied by itself three times, gives the original number. For fractions, finding the cube root involves taking the cube root of both the numerator and the denominator separately.

Converting between mixed numbers and improper fractions is a fundamental skill needed for operations involving mixed numbers.

• Mixed Number: Combines a whole number and a fraction (e.g., \(1\frac{3}{5}\)).
• Improper Fraction: A fraction where the numerator is greater than or equal to the denominator (e.g., \(\frac{8}{5}\)).

Revision Table: Key Concepts

Additional Information on Number Forms

Numbers can be represented in different forms like mixed numbers, improper fractions, decimals, etc. Being able to switch between these forms is crucial for solving various mathematical problems, especially those involving roots and powers.

Finding cube roots of perfect cubes (like 8, 27, 64, 125, 216, 343, 512, 729, 1000) is helpful. It is often beneficial to recognize these values or know how to find them through prime factorization.

Prime factorization of 512: \(512 = 2^9 = (2^3)^3 = 8^3\), so \(\sqrt[3]{512} = 8\).

Prime factorization of 125: \(125 = 5^3\), so \(\sqrt[3]{125} = 5\).

Using prime factorization can help in finding cube roots of larger numbers that are perfect cubes.