Simplifying Expressions with Cube Roots

This problem asks us to simplify a mathematical expression involving the cube roots of negative numbers and a fraction, followed by multiplication and division.

The expression is given by: \(\sqrt[3]{{ - 2197}} \times \sqrt[3]{{ - 125}} \div \sqrt[3]{{\frac{{27}}{{512}}}}\)

To solve this, we need to calculate each cube root term separately and then perform the arithmetic operations.

Step 1: Calculate the first cube root

We need to find the cube root of -2197, which is written as \(\sqrt[3]{{-2197}}\). This is the number that, when multiplied by itself three times, results in -2197.

• Since the number inside the cube root is negative, the cube root will also be negative.
• We look for a number whose cube is 2197. Let's test some integers:
• \(10^3 = 1000\)
• \(20^3 = 8000\)
• The number is between 10 and 20. The last digit of 2197 is 7. We know that \(3^3 = 27\), which ends in 7. So, let's try 13.
• \(13^3 = 13 \times 13 \times 13 = 169 \times 13 = 2197\).
• Therefore, \( (-13)^3 = -13 \times -13 \times -13 = 169 \times -13 = -2197 \).

So, \(\sqrt[3]{{-2197}} = -13\).

Step 2: Calculate the second cube root

Next, we find the cube root of -125, which is \(\sqrt[3]{{-125}}\).

• Similar to the first step, the cube root of a negative number is negative.
• We know that \(5^3 = 5 \times 5 \times 5 = 125\).
• Thus, \( (-5)^3 = -5 \times -5 \times -5 = 25 \times -5 = -125 \).

So, \(\sqrt[3]{{-125}} = -5\).

Step 3: Calculate the third cube root

We need to find the cube root of the fraction \(\frac{{27}}{{512}}\), written as \(\sqrt[3]{{\frac{{27}}{{512}}}}\).

• The cube root of a fraction is the cube root of the numerator divided by the cube root of the denominator: \(\sqrt[3]{{\frac{{a}}{{b}}}} = \frac{{\sqrt[3]{{a}}}}{{\sqrt[3]{{b}}}}\).
• Calculate the cube root of the numerator, 27: \(\sqrt[3]{{27}}\). We know that \(3^3 = 27\), so \(\sqrt[3]{{27}} = 3\).
• Calculate the cube root of the denominator, 512: \(\sqrt[3]{{512}}\). We know that \(8^3 = 8 \times 8 \times 8 = 64 \times 8 = 512\), so \(\sqrt[3]{{512}} = 8\).

So, \(\sqrt[3]{{\frac{{27}}{{512}}}} = \frac{{\sqrt[3]{{27}}}}{{\sqrt[3]{{512}}}} = \frac{{3}}{{8}}\).

Step 4: Substitute the values and simplify the expression

Now we substitute the calculated cube root values back into the original expression:

\(\sqrt[3]{{ - 2197}} \times \sqrt[3]{{ - 125}} \div \sqrt[3]{{\frac{{27}}{{512}}}}\) becomes \((-13) \times (-5) \div \frac{{3}}{{8}}\)

• First, perform the multiplication: \((-13) \times (-5)\). The product of two negative numbers is positive. \(13 \times 5 = 65\). So, \((-13) \times (-5) = 65\).
• The expression is now \(65 \div \frac{{3}}{{8}}\).
• Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of \(\frac{{3}}{{8}}\) is \(\frac{{8}}{{3}}\).
• So, \(65 \div \frac{{3}}{{8}} = 65 \times \frac{{8}}{{3}}\).
• Multiply 65 by 8: \(65 \times 8 = (60 + 5) \times 8 = (60 \times 8) + (5 \times 8) = 480 + 40 = 520\).
• The result is \(\frac{{520}}{{3}}\).

The simplified value of the expression is \(\frac{{520}}{{3}}\).

Comparing with the options:

• Option 1: 492/7
• Option 2: 520/3
• Option 3: 554/7
• Option 4: 571/5

Our calculated result, \(\frac{{520}}{{3}}\), matches Option 2.

Therefore, the simplified value of the expression is \(\frac{{520}}{{3}}\).

Revision Table: Key Cube Roots

Additional Information: Cube Roots Concepts

Understanding cube roots is essential for simplifying expressions like the one in this problem. Here are some key concepts:

• Definition: The cube root of a number \(x\) is a number \(y\) such that \(y^3 = x\). It is denoted by \(\sqrt[3]{x}\).
• Cube Roots of Positive Numbers: Every positive number has exactly one positive real cube root. For example, \(\sqrt[3]{64} = 4\) because \(4^3 = 64\).
• Cube Roots of Negative Numbers: Every negative number has exactly one negative real cube root. For example, \(\sqrt[3]{-8} = -2\) because \((-2)^3 = -8\). This is different from square roots, where negative numbers do not have real square roots.
• Cube Root of Zero: The cube root of zero is zero: \(\sqrt[3]{0} = 0\).
• Cube Roots of Fractions: The cube root of a fraction \(\frac{a}{b}\) is the cube root of the numerator divided by the cube root of the denominator: \(\sqrt[3]{{\frac{{a}}{{b}}}} = \frac{{\sqrt[3]{{a}}}}{{\sqrt[3]{{b}}}}\).
• Perfect Cubes: A perfect cube is an integer that is the cube of an integer (e.g., 1, 8, 27, 64, 125, ... and -1, -8, -27, ...). The numbers -2197, -125, 27, and 512 used in this problem are all perfect cubes.

Knowing the cubes of small integers can help quickly identify perfect cubes and their roots, which simplifies solving problems like this one.