Problem 105 - Intermediate Algebra

Question

Choose the correct letter or letters. No pencil is needed, just think your way through these. Which radical(s) simplify to \(-3 ?\) a. \(\sqrt{9}\) b. \(\sqrt{-9}\) c. \(\sqrt[3]{27}\) d. \(\sqrt[3]{-27}\)

Step-by-Step Solution

Answer

Option d: \( \sqrt[3]{-27} \) simplifies to -3.

Step 1: Understand What Simplifying to -3 Means

To simplify a radical to a value of -3 means that when the radical expression is evaluated, the result is -3. We will examine each option to see if their simplified form is -3.

Step 2: Evaluate Option a

We evaluate the expression \( \sqrt{9} \). The square root of 9 is 3, not -3. Therefore, option (a) does not simplify to -3.

Step 3: Evaluate Option b

We evaluate the expression \( \sqrt{-9} \). The square root of a negative number is not a real number; it involves imaginary numbers (\( 3i \) in this case). Thus, option (b) does not simplify to -3.

Step 4: Evaluate Option c

We evaluate the expression \( \sqrt[3]{27} \). The cube root of 27 is 3, not -3. Therefore, option (c) does not simplify to -3.

Step 5: Evaluate Option d

We evaluate the expression \( \sqrt[3]{-27} \). The cube root of -27 is -3, as \( (-3) \times (-3) \times (-3) = -27 \). Therefore, option (d) simplifies to -3.

Key Concepts

Simplifying Radicals

Simplifying radicals is all about making radical expressions easier to understand and work with.
It involves reducing the expression to its lowest form. Imagine you have a square or cube under a radical sign, often called a radical symbol.
The goal is to find the value that, when multiplied by itself a certain number of times, gives you the number inside this symbol.Here are some tips to simplify radicals:

• Find the largest perfect square or cube that divides into the number under the radical.
• Break down the number under the radical into these factors.
• Simplify by removing the perfect square or cube out of the radical.

Remember, the sign of the result matters too. For example, simplifying \( \sqrt{9} \) gives 3 because 3 multiplied by 3 equals 9.
But note, it will never be negative unless explicitly stated in the problem.

Imaginary Numbers

When dealing with radicals, you might encounter something called imaginary numbers.
These come into play when you try to take the square root of a negative number.Ordinarily, you can't do this within the realm of real numbers. So mathematicians created imaginary numbers:

• The imaginary unit is denoted as \( i \).
• It is defined by the property that \( i^2 = -1 \).
• For instance, \( \sqrt{-9} \) equals \( 3i \) because \( 3i \times 3i = -9 \).

This concept is super useful in advanced math and physics, making it easier to solve problems that involve square roots of negative numbers.
The imaginary numbers work in harmony with real numbers forming what you know as complex numbers.

Cube Roots

Cube roots are quite different from square roots, mainly because they deal with cubes rather than squares.
When you take a cube root, you're looking for a number that multiplies by itself three times to produce the number under the radical.Here's what you need to keep in mind:

• Any real number has exactly one real cube root.
• If the number is positive, the cube root is also positive, and vice versa.
• For example, \( \sqrt[3]{27} = 3 \) because \( 3 \times 3 \times 3 = 27 \).
• Similarly, \( \sqrt[3]{-27} = -3 \) since \( (-3) \times (-3) \times (-3) = -27 \).

With cube roots, the negative sign under the cube radical doesn't cause any complex or imaginary numbers to appear.
Instead, it just means that the cube root is a negative number if the original number was negative.