# Use The Product Rule To Simplify The Expression

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```					  Use The Product Rule To Simplify The Expression
Use The Product Rule To Simplify The Expression

We use the product rule to simplify the expression where expression is a combination of
different kind of variables, numbers and operations like addition, subtraction, multiplication
and division.

Now, we discuss how product rule simplify different kind of expressions: Combination of
algebraic and exponential: If expression is a combination of algebraic and exponential
function, then with the help of product rule we can easily solve differentiation of that
expression.

To simplify with exponents, don't feel like you have to work only from the rules for
exponents. It is often simpler to work directly from the definition and meaning of
exponents. For instance:

Simplify x6 × x5
The rules tell me to add the exponents. But I when I started algebra, I had trouble
keeping the rules straight, so I just thought about what exponents mean.

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The " x6 " means "six copies of x multiplied together", and the " x5 " means "five
copies of x multiplied together". So if I multiply those two expressions together, I will
get eleven copies of x multiplied together. That is:

x6 × x5 = (x6)(x5)
= (xxxxxx)(xxxxx)    (6 times, and then 5 times)
= xxxxxxxxxxx        (11 times)
= x11

Thus:

x6 × x5 = x11

Simplify the following expression:

The exponent rules tell me to subtract the exponents. But let's suppose that I've
forgotten the rules again. The " 68 " means I have eight copies of 6 on top; the " 65 "
means I have five copies of 6 underneath.

How many extra 6's do I have, and where are they? I have three extra 6's, and they're
on top. Then:

Note: If you apply the subtraction rule, you'll end up with 53–9 = 5–6, which is
mathematically correct, but is almost certainly not the answer they're looking for.
Whether or not you've been taught about negative exponents, when they say
"simplify", they mean "simplify the expression so it doesn't have any negative or zero
powers". Some students will try to get around this minus-sign problem by arbitrarily
switching the sign to magically get " 56 " on top (rather than below a "1"), but this is
incorrect.

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Examples
Simplify the trigonometric expression:

syms x
simplify(sin(x)^2 + cos(x)^2)
Ans = 1

Simplify the expression:

syms a b c
simplify(exp(c*log(sqrt(a+b))))
ans =
(a + b)^(c/2)

Simplify the expressions from the matrix:

syms x

S = [(x^2 + 5*x + 6)/(x + 2), sqrt(16)];
R = simplify(S)
R=
[ x + 3, 4]

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Thank You

TutorCircle.com

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