Read powers and evaluate expressions in the correct order

Learning goal: Write repeated multiplication with whole-number exponents and evaluate numerical expressions according to grouping and operation order. Use powers in a formula without confusing an exponent with a multiplier.

In 3⁴, three is the base and four is the exponent. The expression means four factors of three: 3 × 3 × 3 × 3 = 81. It does not mean 3 × 4, which is twelve. The exponent counts repeated factors, not repeated additions. For a positive whole-number exponent, writing the expanded product can clarify an unfamiliar power.

A square with side length four units has area 4² = 4 × 4 = 16 square units. A cube with side length two units has volume 2³ = 2 × 2 × 2 = 8 cubic units. The words squared and cubed connect powers with these geometric examples, but exponents can also describe repeated multiplication outside geometry.

The diagram shows sixteen equal unit squares in a four-by-four array and three equal factors in 2 × 2 × 2. The array's entire area is sixteen square units; the four squares along one edge give a length of four units. Confusing an edge count with an area count would make 4² look like four rather than sixteen.

The order of operations determines how a conventional expression is read. Evaluate grouped expressions first, then powers, then multiplication and division at the same priority from left to right, then addition and subtraction at the same priority from left to right. An acronym should not be interpreted as saying multiplication always precedes division or addition always precedes subtraction.

For 5 + 2³ × 4, evaluate 2³ = 8, then 8 × 4 = 32, then add five to get thirty-seven. For (5 + 2)³ × 4, the parentheses instead make seven the base of the cube: 7³ × 4 = 343 × 4 = 1,372. Grouping changes the mathematical structure, so it can change the result dramatically.

Equal-priority operations follow their written order. In 24 ÷ 6 × 2, first divide twenty-four by six to get four, then multiply by two to get eight. Treating 6 × 2 as a denominator would require parentheses: 24 ÷ (6 × 2) = 2. Similarly, 10 − 3 + 2 = 9, while 10 − (3 + 2) = 5.

A whole-number exponent can be one or zero. For any base a, a¹ = a. For a nonzero base, a⁰ = 1. A pattern explains the zero exponent: 3³ = 27, 3² = 9 and 3¹ = 3 decrease by dividing by three, so continuing one step gives 3⁰ = 1. Do not apply this explanation to 0⁰, which requires a separate convention and is not used in this lesson.

Fractional side lengths can appear in formulas even when the exponent is a whole number. For a cube with side s = 1/2 meter, V = s³ gives (1/2)³ = 1/8 cubic meter. The surface-area formula A = 6s² gives 6 × (1/2)² = 6 × 1/4 = 3/2 square meters. The exponent acts on the side length, while six multiplies the area of one face.

Write a substitution before calculating. A = 6(1/2)² makes clear that the entire one-half side length is squared. Computing (6 × 1/2)² would square a different quantity and produce nine instead of one and one half. Parentheses and unit labels protect the intended structure.

Expressions with letters and powers can also be written from words and read in parts. “The cube of s” is written s³, and “six times the square of s” is 6s². The expression 6s² is a product of two factors, the coefficient 6 and the power s²; within s², s is the base and 2 is the exponent. In 5 + 2³ × 4, the whole expression is a sum of two terms, 5 and 2³ × 4, and the second term is itself a product. Naming the parts shows which operation is completed last.

Worked evaluation: 30 − 2² × 3 = 30 − 4 × 3 = 30 − 12 = 18. Subtracting thirty minus two first would ignore the power and product structure.

Worked correction: “4³ is 4 + 4 + 4.” That sum is 3 × 4 = 12. The power 4³ is 4 × 4 × 4 = 64. State whether the notation counts equal addends or equal factors before computing.

Guided structure work: Expand several powers into factors and rewrite repeated products as powers. Compare pairs of expressions that differ only by parentheses. Explain which operation is completed first and why.

Independent performance: Evaluate expressions involving powers and mixed operations, including equal-priority multiplication/division and addition/subtraction. Use a fractional side length in both cube formulas, label square and cubic units and explain the role of each exponent. Create an incorrect evaluation based on confusing factors with addends, then correct it with a model or expanded product.