Problem 33 - A Graphical Approach to College Algebra

Question

Use a calculator to find each root or power. Give as many digits as your display shows. $$\sqrt[3]{-125}$$

Step-by-Step Solution

Answer

The cube root of -125 is -5.

Step 1: Recognize the Expression

The expression involves finding the cube root of \(-125\). We write this as \(\sqrt[3]{-125}\). This operation asks what number, when multiplied by itself twice (totaling three multiplications), equals \(-125\).

Step 2: Identify the Cube Root of -125

Since the expression is \(-125\), and we're taking the cube root, recognize that the cube root of a negative number is also negative. We need a negative number that, when cubed, equals \(-125\).

Step 3: Use the Calculator

Enter \(-125\) into your calculator and use the cube root function (often labelled as \(\sqrt[3]{x}\) or \(\wedge(1/3)\)). If your calculator does not have a cube root function, you can raise \(-125\) to the power of \(\frac{1}{3}\) using the exponentiation function.

Step 4: Read the Result

The calculator should display the result. When you find the cube root of \(-125\), the display should show \(-5\). This indicates that \(\sqrt[3]{-125} = -5\).

Step 5: Validate the Answer

To confirm, multiply \(-5\) by itself three times: \(-5 \, \times \, -5 \, \times \, -5 = -125\). This verifies that \(-5\) is indeed the cube root of \(-125\).

Key Concepts

Understanding Negative Numbers

Negative numbers can be a bit tricky, especially when dealing with roots and powers. A negative number is less than zero, and you often see them represented with a minus sign, such as \(-125\). When you raise a negative number to an odd power, like cubing, the result will also be negative. This is because:

• Multiplying a negative by a negative gives a positive.
• Multiplying again by a negative flips it back to negative.

So, \(-5 \times -5\) gives \(25\), and then \(25 \times -5\) gives \(-125\). Hence, the cube root of a negative number, like \(-125\), will also be negative.

Mastering Calculators for Cube Roots

Calculators are powerful tools, especially for finding roots and powers quickly. When asked to find the cube root of \(-125\), follow these steps:

• Enter the number, \(-125\), into your calculator.
• Look for a cube root function, often displayed as \(\sqrt[3]{x}\) or sometimes a button sequence like \(\wedge(1/3)\).
• If your calculator does not have a cube root function, you can still calculate by raising the number to the \(1/3\) power.

This will give you the result of \(-5\). Always double-check by reversing the operation to ensure the answer is correct.

Exploring Exponentiation

Exponentiation involves raising a number to a certain power. The cube root is essentially the opposite of cubing a number.
In terms of exponentiation:

• Cubing a number means raising it to the power of 3, i.e., \(x^3\).
• The cube root is finding what raised to the third power gives you the original number, expressed as \((x^{1/3})\).

When we apply this to \(-125\), raising \(-5\) to the third power, or \( (-5)^3\), gets us back to \(-125\). This link between exponentiation and roots is key to understanding complex math problems step-by-step.