Question Video: Simplifying Algebraic Expressions with More than one Variable Involving Exponents and Square Roots | Nagwa Question Video: Simplifying Algebraic Expressions with More than one Variable Involving Exponents and Square Roots | Nagwa

Question Video: Simplifying Algebraic Expressions with More than one Variable Involving Exponents and Square Roots Mathematics • Second Year of Secondary School

Join Nagwa Classes

Attend live General Mathematics sessions on Nagwa Classes to learn more about this topic from an expert teacher!

Write √(25𝑎²𝑏⁶) in its simplest form.

01:56

Video Transcript

Write the square root of 25𝑎 squared 𝑏 to the sixth power in its simplest form.

In order to simplify this expression, we’re going to recall that if the 𝑛th root of 𝑎, the 𝑛th root of 𝑏, and the 𝑛th root of 𝑐 are well defined for positive integers 𝑛, then their product is the 𝑛th root of 𝑎𝑏𝑐. We’re going to apply this property in reverse to allow us to write the original expression as the square root of 25 times the square root of 𝑎 squared times the square root of 𝑏 to the sixth power. Now, we know that the square root of 25 is simply equal to five. But what do we do with the square root of 𝑎 squared and the square root of 𝑏 to the sixth power?

Well, since we’re finding the square root, this is an 𝑛th root where 𝑛 is even; it’s two. So we use the following property. If 𝑛 is even and 𝑎 is a real number, then the 𝑛th root of 𝑎 to the 𝑛th power is equal to the absolute value of 𝑎. So we can actually rewrite the square root of 𝑎 squared as the absolute value of 𝑎. To repeat this process for the square root of 𝑏 to the sixth power, we need to rewrite 𝑏 to the sixth power as 𝑏 cubed squared, meaning that the square root of 𝑏 cubed all squared is equal to the absolute value of 𝑏 cubed.

We can now replace each root with the value we found. So the square root of 25 times the square root of 𝑎 squared times the square root of 𝑏 to the sixth power is five times the absolute value of 𝑎 times the absolute value of 𝑏 cubed. Then, since five is by its very nature nonnegative, we can use the rules for multiplying absolute value expressions to rewrite this as the absolute value of five 𝑎𝑏 cubed. And so in its simplest form, the square root of 25𝑎 squared 𝑏 to the sixth power is equal to the absolute value of five 𝑎𝑏 cubed.

Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

  • Interactive Sessions
  • Chat & Messaging
  • Realistic Exam Questions

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy