Problem 232 - Elementary Algebra

Question

Explain the process for determining whether two radicals are like or unlike. Make sure your answer makes sense for radicals containing both numbers and variables.

Step-by-Step Solution

Answer

Radicals are like if their simplified radicands and root indices are identical.

Step 1: Identify the Radicand

The radicand is the number or expression inside the radical sign. For example, in \(\root{3}{8x^{2}}\), the radicand is \(8x^{2}\).

Step 2: Simplify the Radicands

Simplify the radicands as much as possible. This includes factoring numbers and combining like terms for variables. For example, \(8x^2 = 2^3 x^2\).

Step 3: Compare the Radicands

Determine if the simplified radicands are identical. If they are, the radicals are like radicals. Otherwise, they are unlike radicals.

Step 4: Check the Roots

Ensure that the radicals being compared have the same root index (i.e., both are square roots, cube roots, etc.). The radicals must have the same index to be like radicals.

Key Concepts

radicand identification

When working with radicals, the first step is to identify the radicand. The radicand is the number or the expression inside the radical sign (often called the 'root' sign). For example, in the expression \(\root{3}{8x^{2}}\), the radicand is \(8x^{2}\). Identifying the radicand correctly is crucial because it lays the foundation for further steps. By properly recognizing what part of the expression is inside the radical, we can move on to simplifying and comparing with other radicals accurately. Always remember: the radicand is what's under the root symbol.

simplifying radicals

Once you've identified the radicand, the next step is to simplify it as much as possible. This involves breaking down the number into its prime factors and combining like terms for variables. Simplifying helps to easily compare radicals later on.
For example, let’s simplify the radicand \(8x^{2}\):
1. Factor the number: \(8 = 2^{3}\)
2. Now the expression becomes \(2^{3}x^{2}\)
It’s common to simplify radicals to their simplest form, especially when comparing them or performing arithmetic operations involving radicals.

comparing radicands

After simplifying the radicands, the next step is to compare them. Two radicals are considered like radicals if their simplified radicands are identical. If the radicands are different, then the radicals are unlike.
For example, let’s compare \(\root{3}{8x^{2}}\) and \(\root{3}{16x^{2}y}\):
1. Simplify \(8x^{2} = 2^{3}x^{2}\)
2. Simplify \(16x^{2}y = 2^{4}x^{2}y\)
Since \(2^{3}x^{2}\) and \(2^{4}x^{2}y\) are not the same, these radicals are unlike radicals. Always ensure the simplified forms are compared to determine the likeness.

root index

Finally, it's crucial to ensure the radicals being compared have the same root index. The root index indicates which type of root we are dealing with—square root (index 2), cube root (index 3), and so on. For two radicals to be like radicals, they must not only have identical radicands but also the same root index.
For instance, \(\root{3}{8x^{2}}\) (cube root) and \(\root{2}{8x^{2}}\) (square root) would still be unlike radicals even if their radicands were identical because their root indices differ. Always check that both radicals share the same root index to confirm they are like radicals.