Simplifying Expressions | Brilliant Math & Science Wiki (2024)

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.

Contents

  • Combining Like Terms
  • Multiplying and Dividing Monomials
  • Exponents
  • See Also

Combining Like Terms

"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 \( 2xy + 5 xy \)?

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 \(5xy - 3xy\)?

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.

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 \( 2 \). \(_\square\)

\[2 c\] \[1\] \[c^2\] \[0\]

What is \(a - (a - b - a - c + a) - b + c?\)

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.

Combining like terms, we get \[ \begin{align}y^4 + \frac{1}{2}y - 2y^3 - y^4 + 5y^2 + \frac{5}{2}y + 3 &= \left(y^4 - y^4\right) - 2y^3 + 5y^2 + 3y + 3 \\&= -2y^3 + 5y^2 + 3y + 3 .\end{align}\]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 \( \left(2a^3 - 4a^2 + a - 5\right) - \left(2a + 2 - a^3\right) \).

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 \[ \frac{3a - 1}{2} - \frac{a + 2}{4}. \]

We have

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

Multiplying and Dividing Monomials

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

What is \( 2x^3 \times 5x^7 \)?

We have

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

What is \(25xy × 4xy\)?

We have

\[25xy × 4xy = 100x^2y^2.\ _\square\]

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

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:

\[ \begin{align}2 \div 3 &= 2 \times \frac{1}{3} \\x \div y &= x \times \frac{1}{y} \end{align} \]

and

\[ \begin{align}2 \div \frac{1}{3} &= 2 \times 3 \\x \div \frac{1}{y} &= x \times y.\end{align} \]

What is \(100xy ÷ 10xy\)?

We have

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

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

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 \( \dfrac{2a^5}{b^2} \div \dfrac{7a^3}{b} \)?

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

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

We have

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

Exponents

Main Article: Exponents

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

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

We have

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

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

We have

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

\(36\) \(7\) \(6\) \(42\) \(49\)

\[ \large\color{green} 6^{\color{blue} 6} + \color{green} 6^{\color{blue} 6} + \color{green} 6^{\color{blue} 6} + \color{green} 6^{\color{blue} 6} + \color{green} 6^{\color{blue} 6} + \color{green} 6^{\color{blue} 6} = \color{green}6^ {\color{brown}a} \]

If \(\color{brown} a\) satisfies the equation above, what is the value of \(\color{brown} a\)?

See Also

  • Fractions
  • Rationalizing Denominators
  • Solving Equations
Simplifying Expressions | Brilliant Math & Science Wiki (2024)

FAQs

What is the rule for simplifying expressions? ›

The general rule to simplify expressions is PEMDAS - stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

How do you simplify math expressions? ›

To simplify expressions, one must combine all like terms and solve all specified brackets, if any, until they are left with unlike terms that cannot be further reduced in the simplified expression. As a result of simplify algebraic expressions, the resulting value is that mathematical expression's final product.

What grade do you learn simplifying algebraic expressions? ›

Here you will learn about simplifying expressions, including using the distributive property and combining like terms. Students will first learn about simplifying expressions as part of expressions and equations in 6th grade.

Does the order matter when simplifying expressions? ›

Like with any problem, you'll need to follow the order of operations when simplifying an algebraic expression. The order of operations is a rule that tells you the correct order for performing calculations.

What comes first when simplifying? ›

The order of operations can be remembered by the acronym PEMDAS, which stands for: parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right. First, simplify what is in parentheses. Then, do any exponents.

What is the formula for simplify? ›

Rules of Simplification and Approximation
TopicRules
Exponent of Zeroa0=1
Quotient Ruleaman=am–n
Negative Exponenta−n=1an 1a−n=an (ab)−n=(ba)n
Exponent of Onea0=1
2 more rows
May 3, 2023

What are the basic rules of simplification? ›

The basics of simplification are to use the 'BODMAS' rule. BODMAS stands for: brackets, order of power, division, multiplication, addition, and subtraction. Ans. A vinculum is a horizontal line used in a mathematical expression; the line indicates that the following expression is grouped together.

What is the simplifying strategy in math? ›

• Compensation: a simplifying strategy where students. add or subtract the same amount to or from both. numbers to create an equivalent but easier problem, e.g., 610-290 = 620-300 = 320.

What is the PEMDAS rule? ›

What Does PEMDAS Mean? PEMDAS is an acronym for the words parenthesis, exponents, multiplication, division, addition, subtraction. Given two or more operations in a single expression, the order of the letters in PEMDAS tells you what to calculate first, second, third and so on, until the calculation is complete.

What is an example of an expression in math? ›

An expression or algebraic expression is any mathematical statement which consists of numbers, variables and an arithmetic operation between them. For example, 4m + 5 is an expression where 4m and 5 are the terms and m is the variable of the given expression separated by the arithmetic sign +.

What is the difference between an expression and a polynomial? ›

Are algebraic expressions polynomials? No, not all algebraic expressions are polynomials. But all polynomials are algebraic expressions. The difference is polynomials include only variables and coefficients with mathematical operations(+, -, ×) but algebraic expressions include irrational numbers in the powers as well.

What are the rules for simplifying expressions? ›

When we simplify an expression we operate in the following order:
  1. Simplify the expressions inside parentheses, brackets, braces and fractions bars.
  2. Evaluate all powers.
  3. Do all multiplications and division from left to right.
  4. Do all addition and subtractions from left to right.

How do you simplify hard algebra? ›

To simplify expressions first expand any brackets, next multiply or divide any terms and use the laws of indices if necessary, then collect like terms by adding or subtracting and finally rewrite the expression.

Is 7th grade math hard? ›

7th-grade math can be tough as it introduces you to complex Algebraic-thinking concepts. The difficulty of 7th-grade math depends on factors like – your basic math skills, attention span, and practice skills.

What is the basic rule for simplification? ›

The basics of simplification are to use the 'BODMAS' rule. BODMAS stands for: brackets, order of power, division, multiplication, addition, and subtraction. Ans. A vinculum is a horizontal line used in a mathematical expression; the line indicates that the following expression is grouped together.

What is the rule for math expressions? ›

Terms in an expression are separated by addition and subtraction. Operations could be addition, subtraction, multiplication, or division. An arithmetic expression contains only constant terms separated by operations. An algebraic expression can also contain variables and coefficients.

What is the product rule to simplify expressions? ›

Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the product rule of exponents.

What are the rules for simplifying numerical expressions? ›

Simplification of Numerical Expressions

To simplify a numerical expression that has two or more operations, we perform the BODMAS rule. In this rule, we have to solve operations like Brackets, order of powers or roots, Division first, followed by Multiplication, Addition and then Subtraction.

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