Simplify each radical expression. Use absolute value bars where they are needed. √(-64 a^3 b^6)

Name: Problem 72, Chapter 9: Algebra 2
Uploaded: 2019-08-26T04:19:50-08:00
Duration: 3 min 44 s
Channel: Daniel Pezzi
Description: Simplify each radical expression. Use absolute value bars where they are needed. √(-64 a^3 b^6)

Simplify each radical expression. Use absolute value bars...

Key Concepts

Odd Root Characteristics

When working with odd-index radicals, such as cube roots, the function is defined for all real numbers, including negatives. This means that the cube root of a negative number is also negative, eliminating the need for absolute value bars generally used to ensure non-negativity.

Absolute Value in Radical Simplification

In radical simplification, absolute value bars are used to ensure the non-negative nature of even-index roots when variables could be negative. Although not typically required for odd-index radicals, the concept is important when simplifying expressions to correctly handle even-powered factors under a radical.

Radical Expression Simplification

Simplifying radical expressions involves rewriting the expression under a root sign to extract the largest possible perfect power factors. This process reduces the radical into its simplest form by taking out the factors whose exponents match the index of the radical.

Properties of Exponents

Understanding the laws of exponents is crucial when simplifying radicals. Breaking down expressions with exponents into parts that are multiples of the radical’s index allows for the extraction of perfect power factors, leaving behind any remaining factors inside the radical.

Transcript

00:01 Hello, we need to go through and simplify this expression and write any absolute values if necessary.

00:07 So we have the cube root, a cube root on the outside, so we're looking for three copies of anything on the inside so that we can bring it out.

00:18 So i'm going to break each of these.

00:22 Give me a big cube root.

00:24 I'm going to break each of these into their parts, so it's easy for us to group them together and bring them outside.

00:28 So i'm going to write this negative 1 in a really weird way.

00:33 I'm going to write it as negative 1 times negative 1 times negative 1.

00:38 And if you go through and multiply these out, you see that negative 1 and negative 1 canceled a 1.

00:42 So i left with negative 1, so i haven't changed anything.

00:45 I'm going to write 64 as 2 times 2 times 2.

00:53 Actually, let me do more steps.

00:56 2 times 32.

00:59 Now 32 is the same thing as 2 times 16, but 16 is the same thing as 2 times 8, but 8 is the same thing as 2 times 4, but 4 is the same thing as 2 times 2 times 2.

01:19 What i did there is called a factorization.

01:21 All i'm trying to do is break these complicated things into what's multiplied together to get them.

01:26 I'm sorry that these don't look like twos, but they are.

01:29 So that i can deal with the cube root because everything's multiplied together.

01:34 It's going to be very nice.

01:35 I'm going to write a cubed as it is because this tells me something very nice about it...

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