# Simplify Expressions

Instructions: Use this simplifying of expressions calculator to reduce any valid algebraic NUMERIC expression you provide, showing all the steps. Please type in the expression you want to simplify using PEMDAS rules.

Enter the expression you want to simplify (Ex: sqrt(2/3 + 4/5)+3^2, etc.)

## More about calculator to simplify expressions

This simplify calculator with steps allows you to simplify any valid expression that involves basic operations, including sums, subtraction, multiplications, divisions, fractions, radicals, etc.

All you need to provide is a valid expression involving basic operations. It could be something simple as '1/4 + 1/5', or maybe something more complex like 'sqrt(3)/(3+2^3+5+1/6)'.

Once you provide a valid expression, you need to click on the "Calculate" button, and you all the steps of the simplification calculations will be shown to you.

The calculator will do its best to show meaningful steps for the calculations, and it certainly achieves that for majority of simple expressions.

## How to simplify expressions with multiplication

This question is related another question is how to simplify expressions with sums, and even more interesting, how to simplify expressions that mix sums and multiplications? The answer is simple: PEMDAS

PEMDAS provides a clear rule of what operations have priority to be performed first. Follow these PEMDAS rules:

• First: "P" (that corresponds to "parentheses"). In an algebraic expression, parentheses have priority, always.
• Next: "E" (exponents). After parentheses, priority goes to exponents
• Next: "M" (multiplication). After exponents, priority goes to multiplications
• Next: "D" (division). After multiplications, priority goes to divisions
• Finally: "S" (subtraction). After additions, priority goes to subtractions

These rules will allow you to unequivocally evaluate a compound expression. This calculator will show you the steps of the simplification following the PEMDAS rules of priority

## What are the steps simplifying an expression

• Step 1: Assess whether the expression is well defined. This may not be direct or simple, depending on the complexity of the passed expression
• Step 2: If it is not valid, stop, the process ends. If it is valid, then you use PEMDAS to guide the simplification process
• Step 3: Go simplifying by priority, and take many steps if needed, making sure of following the PEMDAS priority one by one, until the expression cannot be further simplified

## How to simplify expressions with fractions?

It is easy in general to simplify fractions, because the strategy is impossible to miss: you need to find common denominators. For example, the simplest case with 2 fractions, you get:

$\displaystyle \frac{a}{b} \times \frac{c}{d} = \displaystyle \frac{ac}{bd}$

Unfortunately, there are expressions that are much more complicated than simple fractions. But yet, following the right priority of operations, knowing what to operate first, and what next, gives you a clear roadmap to simplify even the most complicated expressions.

## Is this a simplifying radicals calculator?

Yes, that is correct. Calculating radicals or roots is a form a applying an exponent. For example, $$\sqrt 3 = 3^{1/2}$$, which means that the square root of 3 is the same as raising 3 to the 1/2 power (so 1/2 is the exponent).

Now, this calculator will simplify expressions that contains other operations than simply a reduction of radicals. So this calculator is good when simplifying algebraic expressions in general

## Is this a simplify exponents calculator?

Yes. All elementary operations included in PEMDAS are supported by this simplification calculator, including exponents (the "E" in PEMDAS).

Now, when you have exponents mixed with expressions that don't have exponents will yield complex expressions, but that is fine. The worst case scenario is that the expression won't have any further simplifications..

### Example: Calculating a simplification of an expression

Calculate the following: $$\displaystyle \frac{1}{3} + \frac{5}{4} - \frac{5}{6} \times \sqrt{8}$$

Solution: We need to calculate and simplify the following expression: $$\displaystyle \frac{1}{3}+\frac{5}{4}-\frac{5}{6}\cdot\sqrt{8}$$.

The following calculation is obtained:

$$\displaystyle \frac{1}{3}+\frac{5}{4}-\frac{5}{6}\sqrt{8}$$
By simplifying the radical: $$\displaystyle \sqrt{8} = \sqrt{ 2^2 \cdot 2} = 2\sqrt{ 2}$$
$$= \,\,$$
$$\displaystyle \frac{1}{3}+\frac{5}{4}-\frac{5}{6}\cdot 2\sqrt{2}$$
Canceling 2 from the denominator of $$\displaystyle -\frac{ 5}{ 6}$$
$$= \,\,$$
$$\displaystyle \frac{1}{3}+\frac{5}{4}-\frac{5}{3}\sqrt{2}$$
Amplifying in order to get the common denominator 12
$$= \,\,$$
$$\displaystyle \frac{1}{3}\cdot\frac{4}{4}+\frac{5}{4}\cdot\frac{3}{3}-\frac{5}{3}\sqrt{2}$$
We need to use the common denominator: 12
$$= \,\,$$
$$\displaystyle \frac{1\cdot 4+5\cdot 3}{12}-\frac{5}{3}\sqrt{2}$$
Expanding each term: $$4+5 \times 3 = 4+15$$
$$= \,\,$$
$$\displaystyle \frac{4+15}{12}-\frac{5}{3}\sqrt{2}$$
Adding up each term in the numerator
$$= \,\,$$
$$\displaystyle \frac{19}{12}-\frac{5}{3}\sqrt{2}$$

which concludes the calculation.

### Example: Simplifying an expression

Calculate the following: $$\displaystyle \left(\frac{1}{3} + \frac{5}{4} - \frac{5}{6}\right)/(2+3 \times \sqrt{8})$$

Solution: We need to calculate and simplify the following expression: $$\displaystyle \frac{\frac{1}{3}+\frac{5}{4}-\frac{5}{6}}{2+3\sqrt{8}}$$.

The following calculation is obtained:

$$\displaystyle \frac{\frac{1}{3}+\frac{5}{4}-\frac{5}{6}}{2+3\sqrt{8}}$$
By simplifying the radical: $$\displaystyle \sqrt{8} = \sqrt{ 2^2 \cdot 2} = 2\sqrt{ 2}$$
$$= \,\,$$
$$\displaystyle \frac{\frac{1}{3}+\frac{5}{4}-\frac{5}{6}}{2+3\cdot 2\sqrt{2}}$$
Reducing the integers that can be multiplied together: $$\displaystyle 3\times2 = 6$$
$$= \,\,$$
$$\displaystyle \frac{\frac{1}{3}+\frac{5}{4}-\frac{5}{6}}{2+6\sqrt{2}}$$
Amplifying in order to get the common denominator 12
$$= \,\,$$
$$\displaystyle \frac{\frac{1}{3}\cdot \frac{4}{4}+\frac{5}{4}\cdot \frac{3}{3}-\frac{5}{6}\cdot \frac{2}{2}}{2+6\sqrt{2}}$$
Finding a common denominator: 12
$$= \,\,$$
$$\displaystyle \frac{\frac{1\cdot 4+5\cdot 3-5\cdot 2}{12}}{2+6\sqrt{2}}$$
Expanding each term in the numerator: $$4+5 \times 3-5 \times 2 = 4+15-10$$
$$= \,\,$$
$$\displaystyle \frac{\frac{4+15-10}{12}}{2+6\sqrt{2}}$$
$$= \,\,$$
$$\displaystyle \frac{\frac{9}{12}}{2+6\sqrt{2}}$$
We can factor out 3 for both the numerator and denominator.
$$= \,\,$$
$$\displaystyle \frac{\frac{3\cdot 3}{3\cdot 4}}{2+6\sqrt{2}}$$
Now we cancel 3 out from the numerator and denominator.
$$= \,\,$$
$$\displaystyle \frac{\frac{3}{4}}{2+6\sqrt{2}}$$

and this concludes the calculation.

### Example: Another simplification of an expression

Calculate $$\displaystyle \frac{1}{\left(\frac{2}{3} \times \frac{6}{5} \right)} + \frac{2}{5}$$.

Solution: We need to calculate and simplify the following expression: $$\displaystyle \frac{1}{\frac{2}{3}\cdot\frac{6}{5}}+\frac{2}{5}$$.

The following calculation is obtained:

$$\displaystyle \frac{1}{\frac{2}{3}\cdot\frac{6}{5}}+\frac{2}{5}$$
We can multiply the terms in the top and bottom, and we get $$\displaystyle\frac{ 2}{ 3} \times \frac{ 6}{ 5}= \frac{ 2 \times 6}{ 3 \times 5}$$
$$= \,\,$$
$$\displaystyle \frac{1}{\frac{2\cdot 6}{3\cdot 5}}+\frac{2}{5}$$
Factoring out the term $$\displaystyle 3$$ in the numerator and denominator of $$\displaystyle \frac{ 2 \times 6}{ 3 \times 5}$$
$$= \,\,$$
$$\displaystyle \frac{1}{\frac{2\cdot 2}{5}}+\frac{2}{5}$$
After simplifying the common factors in the numerator and denominator
$$= \,\,$$
$$\displaystyle \frac{1}{\frac{4}{5}}+\frac{2}{5}$$
Multiplying by 1 preserves the value: $$\displaystyle 1 \times \frac{ 5}{ 4} = \frac{ 5}{ 4}$$
$$= \,\,$$
$$\displaystyle \frac{5}{4}+\frac{2}{5}$$
Amplifying in order to get the common denominator 20
$$= \,\,$$
$$\displaystyle \frac{5}{4}\cdot\frac{5}{5}+\frac{2}{5}\cdot\frac{4}{4}$$
Finding a common denominator: 20
$$= \,\,$$
$$\displaystyle \frac{5\cdot 5+2\cdot 4}{20}$$
Expanding each term: $$5 \times 5+2 \times 4 = 25+8$$
$$= \,\,$$
$$\displaystyle \frac{25+8}{20}$$
Operating the terms in the numerator
$$= \,\,$$
$$\displaystyle \frac{33}{20}$$

which finalizes the calculation.

## Other useful algebra calculators

Naturally, for simplifying a fraction when no other operation is involved demands a lighter approach. You can also use this expression calculator to get the numerical value of a expression, something that could come in handy.

In terms of fraction operations, you can also use this mixed fraction calculator, which is a simple calculator that is not always available in other calculators.