Subtracting and simplifying radicals | Exponent expressions and equations | Algebra I | Khan Academy

TL;DR

Subtracting and simplifying a complex radical expression can be done step-by-step, resulting in a simplified expression involving absolute values.

Transcript

We're asked to subtract all of this craziness over here. And it looks daunting. But if we really just focus, it actually should be pretty straightforward to subtract and simplify this thing. Because right from the get-go, I have 4 times the fourth root of 81x to the fifth. And from that, I want to subtract 2 times the fourth root of 81x to the fift... Read More

Key Insights

• 😑 Subtracting radical expressions involves subtracting coefficients and simplifying terms with the same radical.
• 😑 Prime factorization can be used to simplify expressions involving radicals.
• 👻 Breaking down exponents into multiples of the radical allows for further simplification.
• 😑 The use of absolute values is necessary in the simplified expression to ensure a positive or zero result.

Questions & Answers

Q: How do you subtract and simplify radical expressions step-by-step?

Subtracting and simplifying radical expressions involves first identifying the terms with the same radical and subtracting their coefficients, followed by simplifying what's inside the radical using prime factorization and converting exponents to multiples of the radical. Once simplified, combining like terms and applying the appropriate rules for the operation of radicals is necessary.

Q: Can negative values be included in the simplified expression?

In the simplified expression, the use of the absolute value is necessary to ensure that the result is positive or zero. This is because taking the fourth root or the principal square root of a negative value would result in a complex number, and the expression assumes real numbers.

Q: What is the domain for the simplified expression?

The simplified expression assumes that the domain is x ≥ 0, meaning x must be greater than or equal to zero for the expression to be defined. For negative values of x, the expression would involve complex or imaginary numbers.

Q: Are there any further simplifications possible for the expression?

The expression cannot be further simplified since the terms involve different radicals: the fourth root and the principal square root. If the radicals were the same, additional simplifications could be made.

Summary & Key Takeaways

• The expression involves subtracting and simplifying two terms: 4 times the fourth root of 81x to the fifth and the principal square root of x to the third.

• The terms with the same radical simplify to 2 times the fourth root of 81x to the fifth.

• Further simplification involves expressing 81 as a fourth power and breaking down x to the fifth as x to the fourth times x.