Indices and Index Laws

Learning Intentions

To understand the meaning of an expression in the form $a^{n}$ in terms of repeated multiplication of $a$
To know the meaning of the terms base, index (plural indices) and expanded form
apply the index laws for multiplying terms with the same base
apply the index laws for dividing terms with the same base

Know that multiplication can be written as repeated addition in simple cases
Understand that repeated multiplication of the same factor can be written more efficiently
Be able to multiply whole numbers and algebraic terms
Know that a term can contain numbers and pronumerals
Be able to simplify simple algebraic products such as $x \times x = x^{2}$
Understand that a fraction bar can represent division
Be able to identify factors in a product

Write each expression in expanded form and state its meaning:
a) $3^{4}$
b) $x^{3}$
c) $a^{5}$

Identify the base and index in each expression:
a) $2^{6}$
b) $m^{4}$
c) $7 x^{3}$

Use the index law for multiplication with the same base:
a) $x^{2} \times x^{5}$
b) $a^{3} \times a^{4}$
c) $m \times m^{6}$

Use the index law for multiplication with the same base:
a) $2^{3} \times 2^{4}$
b) $p^{2} \times p \times p^{3}$
c) $y^{5} \times y^{2}$

Use the index law for division with the same base:
a) $x^{7} \div x^{3}$
b) $a^{9} \div a^{2}$
c) $m^{4} \div m$

Use the index laws to simplify:
a) $p^{2} \times p^{5} \div p^{3}$
b) $b^{8} \div b^{2} \times b$
c) $q^{6} \div q^{4} \times q^{3}$

Write each expression in expanded form and state its meaning:
a) $4^{3}$
b) $y^{4}$
c) $b^{6}$

Identify the base and index in each expression:
a) $5^{7}$
b) $n^{3}$
c) $9 k^{2}$

Use the index law for multiplication with the same base:
a) $x^{4} \times x^{3}$
b) $b^{2} \times b^{6}$
c) $t \times t^{5}$

Use the index law for multiplication with the same base:
a) $3^{2} \times 3^{5}$
b) $r^{3} \times r \times r^{2}$
c) $z^{4} \times z^{3}$

Use the index law for division with the same base:
a) $x^{8} \div x^{2}$
b) $b^{7} \div b^{4}$
c) $t^{5} \div t$

Use the index laws to simplify:
a) $r^{3} \times r^{4} \div r^{2}$
b) $c^{9} \div c^{3} \times c^{2}$
c) $w^{7} \div w^{5} \times w^{4}$

Write each expression in expanded form:
a) $2^{5}$
b) $x^{4}$
c) $a^{3}$
d) $m^{6}$
State the base and the index in each expression:
a) $7^{3}$
b) $p^{5}$
c) $4 x^{2}$
d) $9 y^{4}$
Complete each statement:
a) In $a^{n}$ , the base is ______
b) In $a^{n}$ , the index tells how many times ______ is multiplied by itself
c) The expanded form of $x^{3}$ is ______
Use the index law for multiplication with the same base:
a) $x^{2} \times x^{4}$
b) $a^{5} \times a^{3}$
c) $m \times m^{7}$
d) $p^{3} \times p^{2}$
Use the index law for multiplication with the same base:
a) $2^{4} \times 2^{3}$
b) $y^{2} \times y \times y^{5}$
c) $b^{6} \times b^{2}$
d) $q^{3} \times q^{4} \times q$
Use the index law for division with the same base:
a) $x^{9} \div x^{4}$
b) $a^{6} \div a^{2}$
c) $m^{5} \div m$
d) $t^{8} \div t^{3}$
Use the index law for division with the same base:
a) $3^{7} \div 3^{2}$
b) $p^{4} \div p^{4}$
c) $z^{6} \div z^{5}$
d) $k^{9} \div k^{6}$
Simplify each expression using the index laws:
a) $x^{2} \times x^{3} \div x$
b) $a^{7} \div a^{2} \times a^{4}$
c) $m^{3} \times m^{5} \div m^{6}$
d) $p^{8} \div p^{3} \times p^{2}$

Explain why $a^{4}$ means $a \times a \times a \times a$ , and not $4 a$ .
A student says that in $x^{5}$ , the base is $5$ and the index is $x$ . Explain the mistake.
Noah says that $x^{2} \times x^{3} = x^{6}$ because $2 \times 3 = 6$ . Is he correct? Explain.
Explain why $a^{7} \div a^{3} = a^{4}$ .
A student says that $m^{6} \div m^{2} = m^{3}$ because $6 \div 2 = 3$ . Describe the error.

A square has side length $x^{3}$ cm and another factor of $x^{2}$ is multiplied into a calculation for a pattern. Simplify $x^{3} \times x^{2}$ .
A science formula includes $\frac{a^{8}}{a^{5}}$ . Simplify the expression.
A computer process multiplies $b^{4}$ by $b^{7}$ and then divides by $b^{3}$ . Simplify the result.
A student writes the expanded form of $y^{4}$ to help check a pattern rule. Write the expanded form and then simplify $y^{4} \div y^{2}$ .
A design uses $p^{2} \times p^{3} \times p$ tiles in one section. Write the simplified power of $p$ .
A quantity is modelled by $q^{9} \div q^{4} \times q^{2}$ . Simplify the model.

Students may think $a^{n}$ means $a \times n$ instead of repeated multiplication of $a$
Students may confuse the base with the index
Students may think the index tells the value of the term rather than the number of repeated factors
Students may write the expanded form incorrectly by using addition instead of multiplication
Students may multiply indices when multiplying terms with the same base
Students may divide indices when dividing terms with the same base
Students may apply the index laws when the bases are different
Students may forget that a variable with no written index has index $1$