Simplifying Expressions

To simplify a mathematical expression is to represent it in the least complicated form possible. In general the simplest form is one that has used the fundamental properties of numbers, exponents, algebraic rules, etc. to remove any duplication or redundancy from the expression. It is essentially the opposite of expanding an expression (e.g., with the distributive property).

Simplified expressions are significantly easier to work with than those that have not been simplified.

"Like terms" refer to terms whose variables are exactly the same, but may have different coefficients. For example, the terms $2xy$ and $5xy$ are alike as they have the same variable $xy$. The terms $2xy$ and $2x$ are not alike.

Combining like terms refers to adding (or subtracting) like terms together to make just one term.

What is $2xy + 5 xy$?

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Since $2xy$ and $5xy$ are like terms (with a variable of $xy$), we can add their coefficients together to get $2xy + 5xy = (2+5) xy = 7 xy$. $_\square$

What is $5xy - 3xy$?

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Since $5xy$ and $3xy$ are like terms (with a variable of $xy$), we can subtract their coefficients together to get $5xy - 3xy = (5-3) xy = 2xy$. $_\square$

When there are multiple like terms, arrange the terms in order of decreasing degree and simplify.

What is $x^2 + 3 + 2x^2 - 4x + 7$? Simplify terms and state the degree of the polynomial.

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Since $x^2$ and $2x^2$ are like terms (with a variable of $x^2$), we can combine them.
Since $3$ and $7$ are like terms (with a variable of $1$), we can combine them.
The remaining terms are not alike.

Hence, we get $$x^2 + 3 + 2x^2 - 4x + 7 = (1+2) x^2 - 4x + (3+7) = 3x^2 - 4x + 10.$$ The highest degree term is $x^2$, so the polynomial has degree $2$. $_\square$

What is $y^4 + \frac{1}{2}y - 2y^3 - y^4 + 5y^2 + \frac{5}{2}y + 3$? Simplify terms and state the degree of the polynomial.

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Combining like terms, we get $$\begin{aligned} y^4 + \frac{1}{2}y - 2y^3 - y^4 + 5y^2 + \frac{5}{2}y + 3 &= \left(y^4 - y^4\right) - 2y^3 + 5y^2 + 3y + 3 \\ &= -2y^3 + 5y^2 + 3y + 3 . \end{aligned}$$ The highest degree term is $y^3$, so the polynomial has degree $3$. $_\square$

Remember that when adding and subtracting polynomials, the order of operations still applies.

Simplify $\left(2a^3 - 4a^2 + a - 5\right) - \left(2a + 2 - a^3\right)$.

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Distributing the minus sign across the terms in the second set of parentheses, we get

$$2a^3 - 4a^2 + a - 5 - 2a - 2 + a^3.$$

Collecting similar terms and simplifying, the simplified polynomial is

$$\left(2a^3 + a^3\right) - 4a^2 + (a - 2a) - (5 + 2) = 3a^3 - 4a^2 -a -7. \ _\square$$

When adding and subtracting polynomials that are in fractional form, start by finding the common denominator of each term.

Simplify $$\frac{3a - 1}{2} - \frac{a + 2}{4}.$$

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We have

$$\begin{aligned} \frac{3a - 1}{2} - \frac{a + 2}{4} &= \left( \frac{3a - 1}{2} \times \frac{2}{2} \right) - \frac{a + 2}{4} \\ &= \frac{(6a - 2) - (a+2)}{4} \\ &= \frac{5a - 4}{4}. \ _\square \end{aligned}$$

You can multiply constants with constants, and variables with variables, then apply the laws of exponents.

What is $2x^3 \times 5x^7$?

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We have

$$2x^3 \times 5x^7 = (2 \times 5) \times \left(x^3 \times x^7\right) = 10x^{10}. \ _\square$$

What is $25xy × 4xy$?

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We have

$$25xy × 4xy = 100x^2y^2.\ _\square$$

What is $3ab^2 \times \left(-2a^4b^5\right)$?

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We have

$$3ab^2 \times (-2a^4b^5) = (3 \times -2) \times \left(a^1 \times a^4\right) \times \left(b^2 \times b^5\right) = -6a^5b^7. \ _\square$$

When dividing, you can convert division to multiplication with variables, just as you would do with constants. For example:

$$\begin{aligned} 2 \div 3 &= 2 \times \frac{1}{3} \\ x \div y &= x \times \frac{1}{y} \end{aligned}$$

and

$$\begin{aligned} 2 \div \frac{1}{3} &= 2 \times 3 \\ x \div \frac{1}{y} &= x \times y. \end{aligned}$$

What is $100xy ÷ 10xy$?

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We have

$$100xy ÷ 10xy = \dfrac{100xy}{10xy}= 10.\ _\square$$

What is $8x^3y \div 4xy$?

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We have

$$8x^3y \div 4xy = 8x^3y \times \frac{1}{4xy} = \frac{8}{4} \times \frac{x^3y}{xy} = 2x^2. \ _\square$$

What is $\dfrac{2a^5}{b^2} \div \dfrac{7a^3}{b}$?

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We have

$$\frac{2a^5}{b^2} \div \frac{7a^3}{b} = \frac{2a^5}{b^2} \times \frac{b}{7a^3} = \frac{2}{7} \times \frac{a^5b}{a^3b^2} = \frac{2}{7} \times \frac{a^2}{b} = \frac{2a^2}{7b}. \ _\square$$

Here are a few examples mixing multiplication and division. When doing these types of problems, use your knowledge of order of operations and solve parentheses and exponents first. Convert division to multiplication just as you did above, and remember to multiply constants with constants and variables with variables.

What is $\displaystyle 2x^2y^3 \times \left(3x^3y\right)^2 \div xy^6$?

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We have

$$2x^2y^3 \times \left(3x^3y\right)^2 \div xy^6 = 2x^2y^3 \times 9x^6y^2 \times \frac{1}{xy^6} = (2 \times 9) \times \frac{x^8y^5}{xy^{6}} = \frac{18x^7}{y}. \ _\square$$

What is $\left(-2a^2b^3\right)^2 \div \left(a^3b\right)^2 \times 3a^5b$?

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We have

$$\left(-2a^2b^3\right)^2 \div \left(a^3b\right)^2 \times 3a^5b = 4a^4b^6 \times \frac{1}{a^6b^2} \times 3a^5b = 12a^3b^5. \ _\square$$

Main Article: Exponents

To simplify exponents, we follow the rules of exponents to combine all terms that can be merged.

Simplify $\left(3x^2x^4\right)^2$.

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We have

$$\begin{aligned} \left(3x^2x^4\right)^2 &= \left(3x^6\right)^2 \\ &= 3^2 x^{6\cdot 2} \\ &= 9 x^{12}. \ _\square \end{aligned}$$

Simplify $\dfrac{{(5x^3y^4)}^2 × {(4x^4y)}^3}{{(2x^6y^3)}^6}$.

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We have

$$\begin{aligned} \dfrac{{(5x^3y^4)}^2 × {(4x^4y)}^3}{{(2x^6y^3)}^6} & = \dfrac{25x^6y^8 × 64x^{12}y^3}{64x^{36}y^{18}}\\ & = \dfrac{1600x^{18}y^{11}}{64x^{36}y^{18}}\\ & = \dfrac{25}{x^{18}y^7}. \ _\square \end{aligned}$$

• Fractions
• Rationalizing Denominators
• Solving Equations