Write each expression as a single radical. Then simplify your answer. ·√(√(243(x+1)))

Name: Problem 135, Chapter 0: College Algebra: Real Mathematics, Real People
Uploaded: 2021-03-01T19:09:22-08:00
Duration: 2 min 30 s
Channel: Julie Silva
Description: Write each expression as a single radical. Then simplify your answer. ·√(√(243(x+1)))

Write each expression as a single radical. Then simplify...

Key Concepts

Nested Radicals

Nested radicals occur when one radical is placed within another. Understanding how to convert each radical into an expression with fractional exponents allows one to combine and simplify multiple layers of radicals into a single radical expression.

Rational Exponents

Using rational exponents converts radical notation into an exponential form where the index of the radical becomes the denominator. This makes it easier to apply exponent rules such as the product rule and to simplify the expression by combining like terms.

Exponent Rules

The rules of exponents, including the product rule and power rule, enable transformation of expressions involving powers. By rewriting nested radicals as a single exponent (for example, expressing a square root of a square root as raising to the 1/4 power), one can systematically simplify the expression.

Prime Factorization in Radicals

Breaking down numeric factors into their prime factors helps identify perfect power components that can be taken out of the radical. This technique is essential when simplifying expressions into a more compact form, as it allows extraction of factors whose exponents match the radical's index.

Transcript

00:01 Problem, we're first being asked to rewrite the given expression by just using one radical.

00:05 So to do that, let's remember our rule to go from radical form to rational exponent form, because that's going to help us.

00:11 So that says if we have a, or sorry, let me get the radical part first, that means if we have the n -through of a to the m, using rational exponents, that can be rewritten as a to the m over n power.

00:24 So let's start by rewriting just this first radical using rational exponents.

00:30 So i'm going to bring down the square root.

00:32 So now our base will be this entire radican, the 243 times x plus 1.

00:40 We'll almost use that as a bracket.

00:42 Well, because this entire term is being raised to the first power, that numerator will be one, and the index is two.

00:48 So it'll be raised to the one -half power.

00:50 So now we can do this again.

00:52 We're going to treat this like our base.

00:54 So our base, or our a, is this term 243 times x plus 1, all being raised to the 1 half power.

01:04 But then we're taking this, and now we have to figure out its new exponent.

01:10 Well, this whole expression is being raised to the first, so it'll be 1, and the index of this second radical is 2.

01:17 So now we can use our power rule to only have one exponent.

01:22 That tells us to multiply these two exponents...

Please give Ace some feedback