1976 squareroot (2a + b )^ 1976

[hersheykisses1234]

this is my problem I need help with

 

[hersheykisses1234]

I didnt know how to do the squareroot sign ins behind the 1st 1976

 

[mmm4444bot]

Does the given expression look like this?


\(\displaystyle \sqrt[1976]{(2a + b)^{1976}}\)

Also, what is the instruction that comes with this exercise?

 

[karipeters]

Simplify using absolute value notation when necessary. If root can not be simplified state this. Show work.
1976 squareroot (2a + b )^ 1976. I'm sure that's the question that was needed answered.

 

[mmm4444bot]

1976 squareroot (2a + b )^ 1976. I'm sure that's the question that was needed answered.

Your typing above looks like \(\displaystyle 1976 \cdot \sqrt{(2a + b)^{1976}\)

But, your comments make it seem like you're thinking of the leading 1976 as the radical's index, and not just a factor outside the radical.

If you think that the index is 1976, then do not call the radical a "square root" because it's not.

If you think that the index is 2, then how would you simplify the radical?

I'm thinking that the given expression is shown in my first reply. The radical is a principal 1976th root of a 1976th power.

 

[nyc_function]

Radicals

Does the given expression look like this?


\(\displaystyle \sqrt[1976]{(2a + b)^{1976}}\)

Also, what is the instruction that comes with this exercise?

Are you trying to write the above without using a radical? Is this what you are trying to do?


If that is the case, then we write it as [(2a + b)^(1976)]^(1/1976).


A very strange math question....

 

[karipeters]

Does the given expression look like this?


\(\displaystyle \sqrt[1976]{(2a + b)^{1976}}\)

Also, what is the instruction that comes with this exercise?

How you have written the expression is correct. The instructions say:Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.

 

[mmm4444bot]

Are you trying to write the above without using a radical?

True, the OP did not post the instruction(s) that come with this exercise, but I'm convinced that we're supposed to "simplify the radical".

If [switching to exponential form] is the case, then we [express the radical] as [(2a + b)^(1976)]^(1/1976).

Yup, and that's one approach to simplifying, because there is a property for raising a power to another power. When we simplify the exponential expression above, using the property, we get the base.

In other words, the answer is 2a + b.

But, there's also a property for radicals involving principal nth roots of nth powers.

A very strange math question

I believe that the question is designed with the following property in mind.

When the index of a radical is a Whole number n>1, then the radical notation is a symbol that represents one value, namely the principal nth root of the radicand.

But, if the radicand itself is an nth power as well, then the principal nth root of the nth power is simply the power's base.

\(\displaystyle \sqrt[2]{x^2}\) simplifies to \(\displaystyle x\)

\(\displaystyle \sqrt[3]{x^3}\) simplifies to \(\displaystyle x\)

\(\displaystyle \sqrt[4]{x^4}\) simplifies to \(\displaystyle x\)

\(\displaystyle \sqrt[1976]{x^{1976}}\) simplifies to \(\displaystyle x\)

\(\displaystyle \sqrt[n]{x^n}\) simplifies to \(\displaystyle x\)

 

[mmm4444bot]

How you have written the expression is correct.

Okay. So, we need to text it using a different convention because "1976 squareroot" does not make sense for a 1,976th root.

How about: root[1976]((2a + b)^1976)

That's how many software programs do it.

The instructions say:Simplify. Remember to use absolute-value notation when necessary.

There is no need for absolute-value symbols, when simplifying a principal root. There is only one principal root.

You may be thinking of something like:

Find all values of y in the equation y = root[1976]({2a + b}^1976)

In this case, there are two solutions to the equation, the principal 1,976th root (2a+b) and its opposite (-2a-b). This happens when the radical's index is an even number.

 

[lookagain]

Yup, and that's one approach to simplifying, because there is a property for raising a power to another power. When we simplify the exponential expression above, using the property, we get the base.

In other words, the answer is 2a + b.
\(\displaystyle It \ is \ \ |2a + b|, \ unless \ the \ instructions \ state \ to \ assume \ \)
\(\displaystyle \ \ \ \ \\ \ that \ the \ radicands \ are \ not \ negative.\)

But, if the radicand itself is an nth power as well, then the principal nth root of the nth power is simply the power's base.

\(\displaystyle \sqrt[2]{x^2}\) simplifies to \(\displaystyle x\)
No, this is the definition of |x|.


\(\displaystyle \sqrt[3]{x^3}\) simplifies to \(\displaystyle x\)


\(\displaystyle \sqrt[4]{x^4}\) simplifies to \(\displaystyle x\)
No, it equals |x|.


\(\displaystyle \sqrt[1976]{x^{1976}}\) simplifies to \(\displaystyle x\)
No, it simplifies to |x|.


\(\displaystyle \sqrt[n]{x^n}\) simplifies to \(\displaystyle x\)
if n is odd, but not if n is even.

Example of an positive even exponent with a negative x-value:


\(\displaystyle \text{Let x = -9}\)


\(\displaystyle \sqrt{(-9)^2} \ = \ \sqrt{81} \ = \ 9 \ \ and\)


\(\displaystyle |-9 | = 9\)




This is in contrast to:


\(\displaystyle (\sqrt{-9})(\sqrt{-9}) \ = \ (3i)(3i) \ = \ 9i^2 \ = \ 9(-1) \ = \ -9\)