Problem 8 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$(\sqrt[3]{-27})^{2}$$

Step-by-Step Solution

Answer

The expression evaluates to 9.

Step 1: Simplify the Cube Root

First, calculate the cube root of \(-27\). We know that a cube root of a number \(x\) is any number that, when cubed, gives \(x\). Here, the cube root of \(-27\) is \(-3\) because \((-3)^3 = -27\).

Step 2: Square the Result

Next, square the result obtained from Step 1. We found that \(\sqrt[3]{-27} = -3\). Now, square \(-3\) to get \((-3)^2 = 9\).

Key Concepts

Cube Roots

A cube root is a number that, when multiplied by itself twice, results in the original number. For example, the cube root of \(-27\) is determined by finding which number equals \(-27\) when cubed. In this case, the cube root of \(-27\) is \(-3\). This is because \(-3 \, \times \, -3 \, \times \, -3 = -27\).
Cube roots can involve negative and positive numbers.

• The cube root of a negative number is always negative because multiplying three negative numbers results in a negative product.
• Cube roots of integers might not always be whole numbers, but here, it is.

It's important to remember that cube roots are different from square roots, which only involve raising a number to the power of 2 before extracting the root.

Exponents

Exponents are a way of expressing repeated multiplication of a number by itself. In simple terms, if you see something like \(a^b\), it means \(a\) multiplied by itself \(b\) times.
In evaluating expressions, exponents play a key role.

• Positive exponents indicate how many times you multiply the base number by itself.
• An exponent of 2 is called 'squaring,' involving multiplying the base by itself once.
• Negative exponents represent reciprocal values, but in this example, our focus is on positive ones.

When simplifying expressions with exponents, follow the order of operations and simplify cube roots first if they appear before squaring the number.

Simplifying Expressions

Simplifying expressions is the process of making them easier to understand and calculate. This usually involves reducing them to their simplest form by applying mathematical rules, such as those involving cube roots and exponents. The goal is to make calculations more straightforward and less error-prone.

• First, assess any roots, like cube roots, and simplify them.
• Next, apply exponents to any numbers resulting from the earlier step.

In our example, this means calculating \( \sqrt[3]{-27} \) to yield \(-3\), then squaring that result to conclude with 9. Remember, simplifying doesn't mean changing the expression's value—just stripping it down to a more manageable form. This method is crucial in avoiding complexity when evaluating mathematical expressions.