1. ## Basic PEMDAS question

The book I am using me seems to contradict itself

P- Parenthesis First
E- Exponents and roots second
M - multiplication third from left to right
D- division third from left to right (do both multiplication and division together)
S-subtraction last

However if we are to abide by the order of operations stated above, the book seems to contradict itself in the following example

1. $\displaystyle \sqrt{\frac{4(10+6)}{10+3(5)}}$

Parentheses
First:
$\displaystyle \sqrt{\frac{4(16)}{10+15}}$

Exponents and ROOTS second:Now here is where im lost. The whole expression in this case is underneath a root yet instead of taking the squared root of this whole expression the author precedes to add the numbers together to produce $\displaystyle \sqrt{\frac{64}{25}}$ and then ends simplifying the expression by considering the roots, no where near the second order in the operation $\displaystyle \sqrt{\frac{64}{25}}=\sqrt{\frac{8}{5}}$ as indicated by PEMDAS, instead of exponents/roots coming second it comes last in this case.

2. Originally Posted by allyourbass2212
The book I am using me seems to contradict itself

P- Parenthesis First
E- Exponents and roots second
M - multiplication third from left to right
D- division third from left to right (do both multiplication and division together)
S-subtraction last

However if we are to abide by the order of operations stated above, the book seems to contradict itself in the following example

1. $\displaystyle \sqrt{\frac{4(10+6)}{10+3(5)}}$

Parentheses First:
$\displaystyle \sqrt{\frac{4(16)}{10+15}}$

Exponents and ROOTS second:Now here is where im lost. The whole expression in this case is underneath a root yet instead of taking the squared root of this whole expression the author precedes to add the numbers together to produce $\displaystyle \sqrt{\frac{64}{25}}$ and then ends simplifying the expression by considering the roots, no where near the second order in the operation $\displaystyle \sqrt{\frac{64}{25}}=\sqrt{\frac{8}{5}}$ as indicated by PEMDAS, instead of exponents/roots coming second it comes last in this case.
PEMDAS is a guidline to aid early algeraists in understanding the importance of the fundamental axioms of equality, association, and commutation. Intuitively you know that the root cannot be taken until you have simplfied the sum in the denominator. So, do that and then take the root. The importatant the to understand her is that there are other laws (like $\displaystyle \sqrt{10+5}\neq\sqrt{10}+\sqrt{5}$) that must be adhered to as well. PEMDAS is only absolute if no other laws are being compromised.

3. Originally Posted by allyourbass2212
Exponents and ROOTS second:Now here is where im lost. The whole expression in this case is underneath a root yet instead of taking the squared root of this whole expression the author precedes to add the numbers together to produce $\displaystyle \sqrt{\frac{64}{25}}$ and then ends simplifying the expression by considering the roots, no where near the second order in the operation $\displaystyle \sqrt{\frac{64}{25}}=\sqrt{\frac{8}{5}}$ as indicated by PEMDAS, instead of exponents/roots coming second it comes last in this case.
In problems like those I pretend that symbols like the square root and the absolute value have hidden parentheses, and as such, if there is an expression inside the square root or absolute value I simplify everything inside first before taking the square root or absolute value.

In other words, you can think of this:
$\displaystyle \sqrt{\frac{4(10+6)}{10+3(5)}}$
as this:
$\displaystyle \sqrt{\left(\frac{4(10+6)}{10+3(5)}\right)}$
(But as it is, I don't actually write the extra parentheses.) Then simplify everything inside the square root before taking the square root.

01