Simplifying Algebraic Expressions

To simplify algebraic expression (or to write in the simplest form) means to rewrite it in such way, that it is easier to read and understand.

Algebraic expressions are simplified in the same way as number expressions (using same properties and order of operations), except that in most cases you will end up with algebraic expression, not number.

So, what actions are needed to simplify the algebraic expression?

Combining like terms, using distributive property of multiplication
Performing basic operations (if possible)
Simplifying exponents

In the future, we will discover more rules for simplifying, so you can simplify more complex expressions.

Note, that term "simplify" is somewhat controversial. Sometimes you can easily write simplified expression, but sometimes it is quite hard to decide what form to choose. Evertyhing is up to you (or your teacher)!

For example, $$$\frac{{1}}{\sqrt{{{x}}}}$$$ and $$$\frac{\sqrt{{{x}}}}{{x}}$$$ are the same expressions (just multiply numerator and denominator of the first fraction by $$$\sqrt{{{x}}}$$$ to get the second). But what is simpler?

As for me first fraction is simpler, but many teachers require to write expression with irrationalities in the denominator. For them, only the second fraction is applicable.

Now, let's go through a couple of examples.

We start from combining like terms.

Example 1. Simplify $$${2}{x}+{1}-{4}{x}+{5}$$$.

Using commutative property of addition, we can rewrite expression as $$${2}{x}-{4}{x}+{1}+{5}$$$.

Now add 1 and 5: $$${2}{x}-{4}{x}+{6}$$$.

Use distributive property of multiplication: $$${\left({2}-{4}\right)}{x}+{6}=-{2}{x}+{6}$$$.

We can't simplify further, because there are no like terms.

So, answer is $$${\color{purple}{{-{2}{x}+{6}}}}$$$.

Now, consider simplifying exponents.

Example 2. Simplify the following: $$$\frac{{{2}{{x}}^{{4}}{w}\cdot{{w}}^{{2}}{x}+{3}{w}}}{{w}}$$$.

We start from numerator and use distributive property of multiplication: $$$\frac{{{2}{{x}}^{{4}}{w}\cdot{{w}}^{{2}}{x}+{3}{w}}}{{w}}=\frac{{{\left({2}{{x}}^{{4}}\cdot{{w}}^{{2}}{x}+{3}\right)}{w}}}{{w}}$$$.

We can cancel out $$${w}$$$, using rule for subtracting exponents: $$$\frac{{w}}{{w}}={{w}}^{{{1}-{1}}}={{w}}^{{0}}={1}$$$: $$$\frac{{{\left({2}{{x}}^{{4}}\cdot{{w}}^{{2}}{x}+{3}\right)}{w}}}{{w}}={\left({2}{{x}}^{{4}}\cdot{{w}}^{{2}}{x}+{3}\right)}\cdot\frac{{w}}{{w}}={\left({2}{{x}}^{{4}}\cdot{{w}}^{{2}}{x}+{3}\right)}\cdot{1}$$$.

Now, use identity property of multiplication: $$${\left({2}{{x}}^{{4}}\cdot{{w}}^{{2}}{x}+{3}\right)}\cdot{1}={2}{{x}}^{{4}}\cdot{{w}}^{{2}}{x}+{3}$$$.

Next, use commutative property of multiplication first: $$${2}{{x}}^{{4}}\cdot{x}\cdot{{w}}^{{2}}+{3}$$$.

Finally, use a rule for adding exponents: $$${2}{{x}}^{{4}}\cdot{x}\cdot{{w}}^{{2}}+{3}={2}{{x}}^{{{4}+{1}}}{{w}}^{{2}}+{3}={2}{{x}}^{{5}}{{w}}^{{2}}+{3}$$$.

Expression can't be further simplified, so answer is $$${\color{purple}{{{2}{{x}}^{{5}}{{w}}^{{2}}+{3}}}}$$$.

Admit, that it is much more simpler that initial one.

Now, it is time to exercise.

Exercise 1. Simplify $$${2}+{{x}}^{{2}}-{1}+{3}{{x}}^{{3}}-{5}{{x}}^{{2}}+{3}+{6}{{x}}^{{2}}$$$.

Answer: $$${3}{{x}}^{{3}}+{2}{{x}}^{{2}}+{4}$$$.

Exercise 2. Simplify the following: $$$\frac{{{2}{{y}}^{{2}}{{x}}^{{2}}{{y}}^{{3}}{{x}}^{{3}}+{5}{{x}}^{{3}}{{y}}^{{3}}}}{{{w}\cdot{3}{{y}}^{{3}}}}$$$.

Answer: $$$\frac{{{2}{{x}}^{{5}}{{y}}^{{2}}+{5}{{x}}^{{3}}}}{{{3}{w}}}$$$.

Exercise 3. Rewrite, using positive exponents: $$${{\left({{x}}^{{2}}\cdot{{y}}^{{-{2}}}\cdot{{x}}^{{3}}\right)}}^{{-{3}}}+{\sqrt[{{7}}]{{{\sqrt[{{3}}]{{{{x}}^{{2}}}}}}}}$$$.

Answer: $$$\frac{{{{y}}^{{6}}}}{{{{x}}^{{15}}}}+{{x}}^{{\frac{{2}}{{21}}}}$$$. Hint: use a rule for multiplying exponents and properties of exponents.