Problem 61 - Elementary Algebra

Question

\(\sqrt[3]{-216}\)

Step-by-Step Solution

Answer

-6

Step 1: Identify the given expression

The problem requires finding the cube root of \(-216\).

Step 2: Understand the cube root operation

The cube root operation asks for a number \('a'\) such that when multiplied by itself three times, it results in \(-216\). Mathematically, this can be expressed as \(a^3 = -216\).

Step 3: Determine the cube root

\(-6 \times -6 \times -6 = -216\), therefore \(throot 3{-216} = -6\)

Key Concepts

Cube Roots

Cube roots are a way to find a number that, when multiplied by itself three times, gives the original number. For example, if we have the cube root of 27, we are looking for the number that when used in the expression \( x^3 = 27 \), equals to 27. In this case, the answer is 3 because \(3 \times 3 \times 3 = 27\).

The cube root is often written using the radical symbol like this: \( \sqrt[3]{number} \). Unlike square roots, cube roots can have both positive and negative solutions since multiplying three negative numbers also yields a negative result. This makes cube roots very versatile and useful in mathematics and real-life applications.

Negative Numbers

Negative numbers are values less than zero. They are represented with a minus (-) sign. Understanding negative numbers is important when dealing with various mathematical operations like addition, subtraction, multiplication, and division.

When you multiply or divide two negative numbers, the result is positive because the negative signs cancel each other out. For example, \( -2 \times -3 = 6 \). However, when you multiply an odd number of negative numbers, the result stays negative. For example, \( -3 \times -3 \times -3 = -27 \).

Negative numbers are crucial in solving many algebraic equations and real-world problems, such as calculating debts or temperatures below zero.

Exponents

Exponents are a way to represent repeated multiplication of the same number by itself. The expression \(a^b\) means 'a' multiplied by itself 'b' times. For example, \(2^3\) means \(2 \times 2 \times 2 = 8\).

Exponents are very useful for expressing large numbers in a compact form. They also make it easier to perform calculations and solve equations. One of the rules to remember is that any number raised to the power of zero is one, \(a^0 = 1\).

When combined with negative numbers, exponents can produce interesting results. Take \((-2)^3\), for instance. The exponent 3 means we multiply \(-2 \times -2 \times -2 = -8\). Notice how the answer is negative because the exponent is an odd number. Understanding these rules will help you solve more complex problems effectively.