Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 2: x Marks the Spot

Lesson 2: Arithmetic with Expressions

In this Grade 4 lesson from AoPS Introduction to Algebra, students learn to evaluate algebraic expressions by substituting values for variables, combine like terms such as 5x and 3x, and simplify expressions involving exponents and radicals. The lesson also covers critical rules for working with fractions and powers, including when canceling common factors is and is not valid. Part of Chapter 2 in the AMC 8 and 10 preparation curriculum, this lesson builds foundational skills in arithmetic with expressions that support more advanced algebraic reasoning.

Section 1

Procedure for Combining Like Terms

Property

To combine like terms, first identify the terms that have the same variables and exponents.
Next, rearrange the expression so the like terms are grouped together.
Finally, add or subtract the coefficients of the like terms to simplify the expression.

Examples

To simplify $7a + 4b + 2a$ , we identify $7a$ and $2a$ as like terms and combine them to get $9a + 4b$ .
In the expression $5x^2 + 9 + 2x^2 - 4$ , we combine the $x^2$ terms to get $7x^2$ and the constants to get $5$ . The result is $7x^2 + 5$ .
To simplify $10y + 3y^2 + 2y + 6y^2$ , we combine the $y^2$ terms to get $9y^2$ and the $y$ terms to get $12y$ . The simplified expression is $9y^2 + 12y$ .

Explanation

Combining like terms simplifies an expression by grouping similar items. Just as you would group 3 apples and 4 apples to get 7 apples, you combine terms like $3x$ and $4x$ to get $7x$ .

Section 2

First law of exponents

Property

To multiply two powers with the same base, we add the exponents and leave the base unchanged. In symbols,

a^m \cdot a^n = a^{m+n}

Examples

To multiply two powers with the same base, add their exponents: $x^5 \cdot x^3 = x^{5+3} = x^8$ .

Section 3

Second law of exponents

Property

To divide two powers with the same base, we subtract the smaller exponent from the larger one, and keep the same base.

If the larger exponent occurs in the numerator, put the power in the numerator.

\frac{a^m}{a^n} = a^{m-n} \text{ if } n < m

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Procedure for Combining Like Terms

Property

Examples

To simplify $7a + 4b + 2a$ , we identify $7a$ and $2a$ as like terms and combine them to get $9a + 4b$ .
In the expression $5x^2 + 9 + 2x^2 - 4$ , we combine the $x^2$ terms to get $7x^2$ and the constants to get $5$ . The result is $7x^2 + 5$ .
To simplify $10y + 3y^2 + 2y + 6y^2$ , we combine the $y^2$ terms to get $9y^2$ and the $y$ terms to get $12y$ . The simplified expression is $9y^2 + 12y$ .

Explanation

Combining like terms simplifies an expression by grouping similar items. Just as you would group 3 apples and 4 apples to get 7 apples, you combine terms like $3x$ and $4x$ to get $7x$ .

Section 2

First law of exponents

Property

To multiply two powers with the same base, we add the exponents and leave the base unchanged. In symbols,

a^m \cdot a^n = a^{m+n}

Examples

To multiply two powers with the same base, add their exponents: $x^5 \cdot x^3 = x^{5+3} = x^8$ .

Section 3

Second law of exponents

Property

To divide two powers with the same base, we subtract the smaller exponent from the larger one, and keep the same base.

If the larger exponent occurs in the numerator, put the power in the numerator.

\frac{a^m}{a^n} = a^{m-n} \text{ if } n < m

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Procedure for Combining Like Terms

Property

Examples

To simplify $7a + 4b + 2a$ , we identify $7a$ and $2a$ as like terms and combine them to get $9a + 4b$ .
In the expression $5x^2 + 9 + 2x^2 - 4$ , we combine the $x^2$ terms to get $7x^2$ and the constants to get $5$ . The result is $7x^2 + 5$ .
To simplify $10y + 3y^2 + 2y + 6y^2$ , we combine the $y^2$ terms to get $9y^2$ and the $y$ terms to get $12y$ . The simplified expression is $9y^2 + 12y$ .

Explanation

Combining like terms simplifies an expression by grouping similar items. Just as you would group 3 apples and 4 apples to get 7 apples, you combine terms like $3x$ and $4x$ to get $7x$ .

Section 2

First law of exponents

Property

To multiply two powers with the same base, we add the exponents and leave the base unchanged. In symbols,

a^m \cdot a^n = a^{m+n}

Examples

To multiply two powers with the same base, add their exponents: $x^5 \cdot x^3 = x^{5+3} = x^8$ .

Section 3

Second law of exponents

Property

To divide two powers with the same base, we subtract the smaller exponent from the larger one, and keep the same base.

If the larger exponent occurs in the numerator, put the power in the numerator.

\frac{a^m}{a^n} = a^{m-n} \text{ if } n < m