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Prealgebra / LEVEL 2 · DIFFICULTY 1/5

Powers & Exponent Structure

Reason about repeated multiplication, exponent laws and reciprocal powers.

3 stages · 24 practice problems · two 6-question assessment forms

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MINI QUEST 2.1Powers, Signs & GroupingRead the lesson
MINI QUEST 2.2Products, Quotients & Powers of PowersRead the lesson
MINI QUEST 2.3Zero & Negative ExponentsRead the lesson
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STAGE 2.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Powers, Signs & Grouping

Useful preparation: Signed Numbers & Order of Operations

Goal: Distinguish a negative base from a negative sign outside a power.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

An exponent counts equal factors

For a positive integer n, a to the power n is a product of n copies of a. The exponent does not mean multiply a by n.

\displaystyle a^{3} = a \times a \times a

Group the base explicitly

Parentheses make a negative number the base. Without parentheses, a leading minus is applied after the power.

\displaystyle \left(-4\right)^{2} = 16

;

\displaystyle -4^{2} = -16

Use parity to predict the sign

An even number of negative factors has positive product. An odd number has negative product. Predicting the sign gives a useful check.

\displaystyle \left(-a\right)^{2n} = a^{2n}

WORKED EXAMPLE 1

Evaluate $\displaystyle \left(-3\right)^{4} - 3^{2}$ .

Four negative factors give a positive 81.
The second power is 9; subtract it.

WORKED EXAMPLE 2

A colony triples each hour. It starts with 2 cells. How many after four hours?

Four rounds of tripling multiply by $\displaystyle 3^{4}$ .
$\displaystyle 2 \times 81 = 162$ .

162

Common pitfalls

Possible mix-up: $\displaystyle 5^{3}$ is 15.

$\displaystyle 5^{3} = 5 \times 5 \times 5 = 125$ .

Possible mix-up: $\displaystyle -6^{2}$ is 36.

The power acts before the leading minus: $\displaystyle -6^{2} = -36$ .

Explain it to yourself

Compare $\displaystyle \left(-2\right)^{5}$ and $\displaystyle -2^{5}$ . Why do they agree even though the grouping differs?

Preview the eight practice prompts

Problem 1
Compute $\displaystyle 4^{3}$ .
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Problem 2
Compute $\displaystyle \left(-5\right)^{2}$ .
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Problem 3
Compute $\displaystyle -5^{2}$ , where the minus is outside the power.
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Problem 4
Compute $\displaystyle \left(-2\right)^{5}$ .
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Problem 5
Evaluate $\displaystyle 2^{4} + 3^{2} \times 2$ .
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Problem 6
How many negative integers x satisfy $\displaystyle x^{2} = 49$ ?
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Problem 7New context
A digital pattern starts with 5 lights. Each stage doubles the number. How many lights are present after six stages?
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Problem 8New context
A puzzle awards $\displaystyle 3^{2}$ points for each of four clues and subtracts $\displaystyle 2^{3}$ points once. What score results?
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Open stage 2.1 in the student workspace →

STAGE 2.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Products, Quotients & Powers of Powers

Useful preparation: Powers, Signs & Grouping

Goal: Count factors to justify exponent rules for a common base.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Products join factor lists

When powers have the same base, multiplication joins their repeated factors, so exponents add. Different bases need other reasoning.

\displaystyle a^{m}\, a^{n} = a^{m+n}

Quotients cancel factors

For a nonzero base, divide powers by canceling equal factors. The difference of the exponents records what remains.

\displaystyle a^{m} \div a^{n} = a^{m-n}

A power of a power repeats a whole group

Taking the nth power of $\displaystyle a^{m}$ creates n groups of m factors; the total count is mn. A product raised to a power repeats each factor.

\displaystyle \left(a^{m}\right)^{n} = a^{mn}

;

\displaystyle \left(ab\right)^{n} = a^{n}\, b^{n}

WORKED EXAMPLE 1

Write $\displaystyle \left(2^{3}\right)^{4} \times 2^{2}$ as 2 to a single power.

The nested power contributes $\displaystyle 3 \times 4 = 12$ factors of 2.
The final product adds 2 more factors.

$\displaystyle 2^{14}$

WORKED EXAMPLE 2

Compute $\displaystyle 5^{6} \div 5^{4}$ .

Cancel four factors of 5 from numerator and denominator.
Two factors remain, giving $\displaystyle 5^{2}$ .

Common pitfalls

Possible mix-up: $\displaystyle a^{3} + a^{4}$ equals $\displaystyle a^{7}$ .

The exponent addition rule is for multiplication, not addition.

Possible mix-up: $\displaystyle \left(a^{2}\right)^{3}$ equals $\displaystyle a^{5}$ .

There are three groups of two factors, so the exponent is 6.

Explain it to yourself

Use repeated factors to explain why exponent multiplication appears for a power of a power.

Preview the eight practice prompts

Problem 1
Write $\displaystyle 3^{4} \times 3^{2} = 3^{n}$ . Find n.
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Problem 2
Write $\displaystyle 7^{8} \div 7^{3} = 7^{n}$ . Find n.
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Problem 3
Write $\displaystyle \left(5^{2}\right)^{4} = 5^{n}$ . Find n.
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Problem 4
Compute $\displaystyle 2^{7} \div 2^{4}$ .
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Problem 5
Write $\displaystyle \left(3^{2} \times 3^{4}\right)^{3} = 3^{n}$ . Find n.
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Problem 6
Is $\displaystyle 2^{3} + 2^{4}$ equal to $\displaystyle 2^{7}$ ? Enter yes or no.
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Problem 7New context
A grid has $\displaystyle 2^{4}$ rows and $\displaystyle 2^{6}$ columns. If it has $\displaystyle 2^{n}$ cells, what is n?
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Problem 8New context
A drive contains $\displaystyle 3^{9}$ equal files. A package holds $\displaystyle 3^{4}$ files. If the number of full packages is $\displaystyle 3^{n}$ , find n.
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Open stage 2.2 in the student workspace →

STAGE 2.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Zero & Negative Exponents

Useful preparation: Products, Quotients & Powers of Powers · Division, Reciprocals & Zero

Goal: Extend exponent patterns to one and reciprocal values.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

A zero exponent produces the multiplicative identity

For a nonzero base, $\displaystyle \frac{a^{n}}{a^{n}} = 1$ . The exponent quotient pattern gives $\displaystyle a^{0}$ , so $\displaystyle a^{0}$ must be 1. This argument does not define $\displaystyle 0^{0}$ .

\displaystyle a^{0} = 1

for

\displaystyle a \ne 0

Stepping down divides by the base

Each step downward in an exponent divides by the base. Continuing below zero produces reciprocal powers, not negative values.

\displaystyle a^{-n} = \frac{1}{a^{n}}

for

\displaystyle a \ne 0

Keep sign and exponent separate

A negative exponent changes reciprocal structure. The sign of the base still determines the sign of an odd or even power.

\displaystyle \left(-2\right)^{-3} = \frac{-1}{8}

WORKED EXAMPLE 1

Evaluate $\displaystyle 3^{-2} + 3^{0}$ .

$\displaystyle 3^{-2} = \frac{1}{9}$ , while $\displaystyle 3^{0} = 1$ .
Combine $\displaystyle \frac{1}{9} + \frac{9}{9}$ .

$\displaystyle \frac{10}{9}$

WORKED EXAMPLE 2

A magnifier rescales a length by $\displaystyle 2^{-3}$ . What fraction of the original length remains?

The exponent indicates the reciprocal of $\displaystyle 2^{3}$ .
Subtract exponents first: $\displaystyle \frac{2^{7}}{2^{4}} = 2^{3} = 8$ .

$\displaystyle \frac{1}{8}$

Common pitfalls

Possible mix-up: $\displaystyle a^{0}$ is 0.

For nonzero a, $\displaystyle a^{0}$ is 1.

Possible mix-up: $\displaystyle 4^{-2}$ is −16.

The negative exponent means reciprocal: $\displaystyle \frac{1}{16}$ .

Explain it to yourself

Explain the difference between $\displaystyle -3^{2}$ , $\displaystyle \left(-3\right)^{2}$ and $\displaystyle 3^{-2}$ .

Preview the eight practice prompts

Problem 1
Compute $\displaystyle 12^{0}$ .
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Problem 2
Compute $\displaystyle 2^{-4}$ . Enter a fraction.
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Problem 3
Compute $\displaystyle \left(-3\right)^{-3}$ .
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Problem 4
Compute $\displaystyle 5^{3} \times 5^{-4}$ .
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Problem 5
Compute $\displaystyle \left(\frac{2}{3}\right)^{-2}$ .
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Problem 6
Write $\displaystyle 10^{-3} \div 10^{-5} = 10^{n}$ . Find n.
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Problem 7New context
A map scale multiplies every length by $\displaystyle 3^{-2}$ . A road measures 72 units before scaling. How long is it afterward?
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Problem 8New context
A sensor multiplies a signal by $\displaystyle 2^{-3}$ , then another circuit multiplies it by $\displaystyle 2^{5}$ . What is the combined numerical multiplier?
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Open stage 2.3 in the student workspace →

Browse another level →

Powers & Exponent Structure

Powers, Signs & Grouping

An exponent counts equal factors

Group the base explicitly

Use parity to predict the sign

Evaluate (−3)4−32\displaystyle \left(-3\right)^{4} - 3^{2}(−3)4−32.

A colony triples each hour. It starts with 2 cells. How many after four hours?

Common pitfalls

Explain it to yourself

Products, Quotients & Powers of Powers

Products join factor lists

Quotients cancel factors

A power of a power repeats a whole group

Write (23)4×22\displaystyle \left(2^{3}\right)^{4} \times 2^{2}(23)4×22 as 2 to a single power.

Compute 56÷54\displaystyle 5^{6} \div 5^{4}56÷54.

Common pitfalls

Explain it to yourself

Zero & Negative Exponents

A zero exponent produces the multiplicative identity

Stepping down divides by the base

Keep sign and exponent separate

Evaluate 3−2+30\displaystyle 3^{-2} + 3^{0}3−2+30.

A magnifier rescales a length by 2−3\displaystyle 2^{-3}2−3. What fraction of the original length remains?

Common pitfalls

Explain it to yourself

Evaluate $\displaystyle \left(-3\right)^{4} - 3^{2}$ .

Write $\displaystyle \left(2^{3}\right)^{4} \times 2^{2}$ as 2 to a single power.

Compute $\displaystyle 5^{6} \div 5^{4}$ .

Evaluate $\displaystyle 3^{-2} + 3^{0}$ .

A magnifier rescales a length by $\displaystyle 2^{-3}$ . What fraction of the original length remains?