Powers Examples: Solved Problems and Exercises

Calculate the power

Example 1:

2^{3}

2^{3} = 2 \cdot 2 \cdot 2 = 8

Example 2:

5^{2}

5^{2} = 5 \cdot 5 = 25

Example 3:

1 0^{4}

1 0^{4} = 10 \cdot 10 \cdot 10 \cdot 10 = 10000

Example 4:

3^{4}

3^{4} = 3 \cdot 3 \cdot 3 \cdot 3 = 81

Example 5:

0. 1^{3}

0. 1^{3} = 0.1 \cdot 0.1 \cdot 0.1 = 0.001

Examples with rules of powers

Multiplying powers with the same base

Formula: $a^{m} \cdot a^{n} = a^{m + n}$

Example 1:

2^{3} \cdot 2^{2}

2^{3} \cdot 2^{2} = 2^{3 + 2} = 2^{5} = 32

Example 2:

5^{1} \cdot 5^{3}

5^{1} \cdot 5^{3} = 5^{1 + 3} = 5^{4} = 625

Example 3:

1 0^{2} \cdot 1 0^{4}

1 0^{2} \cdot 1 0^{4} = 1 0^{2 + 4} = 1 0^{6} = 1000000

Dividing powers with the same base

Formula: $a^{m} : a^{n} = a^{m - n}$

Example 1:

2^{5} : 2^{3}

2^{5} : 2^{3} = 2^{5 - 3} = 2^{2} = 4

Example 2:

1 0^{6} : 1 0^{2}

1 0^{6} : 1 0^{2} = 1 0^{6 - 2} = 1 0^{4} = 10000

Example 3:

3^{4} : 3^{4}

3^{4} : 3^{4} = 3^{4 - 4} = 3^{0} = 1

Raising a power to a power

Formula: $(a^{m})^{n} = a^{m \cdot n}$

Example 1:

(2^{3})^{2}

(2^{3})^{2} = 2^{3 \cdot 2} = 2^{6} = 64

Example 2:

(5^{2})^{3}

(5^{2})^{3} = 5^{2 \cdot 3} = 5^{6} = 15625

Example 3:

(1 0^{2})^{4}

(1 0^{2})^{4} = 1 0^{2 \cdot 4} = 1 0^{8} = 100000000

Examples with negative exponents

Converting to fraction

Formula: $a^{- n} = \frac{1}{a ^{n}}$

Example 1:

2^{- 3}

2^{- 3} = \frac{1}{2 ^{3}} = \frac{1}{8}

Example 2:

5^{- 2}

5^{- 2} = \frac{1}{5 ^{2}} = \frac{1}{25}

Example 3:

1 0^{- 4}

1 0^{- 4} = \frac{1}{1 0 ^{4}} = \frac{1}{10000}

Mixed operations

Example 1:

2^{- 3} \cdot 2^{2}

2^{- 3} \cdot 2^{2} = 2^{- 3 + 2} = 2^{- 1} = \frac{1}{2}

Example 2:

\frac{1 0 ^{3}}{1 0 ^{- 2}}

\frac{1 0 ^{3}}{1 0 ^{- 2}} = 1 0^{3 - (- 2)} = 1 0^{5} = 100000

Examples with decimal numbers

Squaring decimals

Example 1:

0. 5^{2}

0. 5^{2} = 0.5 \cdot 0.5 = 0.25

Example 2:

0. 2^{2}

0. 2^{2} = 0.2 \cdot 0.2 = 0.04

Example 3:

1. 5^{2}

1. 5^{2} = 1.5 \cdot 1.5 = 2.25

Decimal exponents

Example 1:

0. 1^{2}

0. 1^{2} = 0.1 \cdot 0.1 = 0.01 = 1 0^{- 2}

Example 2:

0.0 1^{2}

0.0 1^{2} = 0.01 \cdot 0.01 = 0.0001 = 1 0^{- 4}

Examples with negative base

Even exponent

(- 2)^{2} = (- 2) \cdot (- 2) = 4

(positive)

(- 3)^{2} = (- 3) \cdot (- 3) = 9

(positive)

Odd exponent

(- 2)^{3} = (- 2) \cdot (- 2) \cdot (- 2) = - 8

(negative)

(- 3)^{3} = - 27

(negative)

Comparison

Comparison of expressions:

$(- 2)^{2} = (- 2) \cdot (- 2) = 4$ → + (even exponent)
$(- 2)^{3} = (- 2) \cdot (- 2) \cdot (- 2) = - 8$ → - (odd exponent)
$(- 3)^{2} = (- 3) \cdot (- 3) = 9$ → + (even exponent)
$(- 3)^{3} = (- 3) \cdot (- 3) \cdot (- 3) = - 27$ → - (odd exponent)
$(- 4)^{4} = 256$ → + (even exponent)

Equations with powers

Simple equation: $x^{2} = 16$

x^{2} = 16

x = \pm 16

x = \pm 4

Check:

$(- 4)^{2} = 16$ ✓
$4^{2} = 16$ ✓

Equation with exponents: $2^{x} = 32$

2^{x} = 32

2^{x} = 2^{5}

x = 5

Equation with negative exponent: $3^{x} = \frac{1}{9}$

3^{x} = \frac{1}{9}

3^{x} = 3^{- 2}

x = - 2

Practical problems

Problem 1: Area of a square

A square has a side length of $5 \cdot 1 0^{2}$ cm. What is its area?

A = (5 \cdot 1 0^{2})^{2} = 25 \cdot 1 0^{4} = 2.5 \cdot 1 0^{5} cm^{2}

Problem 2: Distance and speed

Light travels at $3 \cdot 1 0^{8}$ m/s. How far does it travel in $2 \cdot 1 0^{2}$ seconds?

d = v \cdot t = (3 \cdot 1 0^{8}) \cdot (2 \cdot 1 0^{2}) = 6 \cdot 1 0^{10} m

Problem 3: Population growth

A town has $1 0^{4}$ inhabitants. The population triples every year. What will the population be after 3 years?

P = 1 0^{4} \cdot 3^{3} = 1 0^{4} \cdot 27 = 2.7 \cdot 1 0^{5} inhabitants

Interactive exercises

Practice with our interactive exercises:

Summary of formulas

Rule: Multiplication → Formula: $a^{m} \cdot a^{n} = a^{m + n}$ → Example: $2^{3} \cdot 2^{2} = 2^{5} = 32$
Rule: Division → Formula: $a^{m} : a^{n} = a^{m - n}$ → Example: $2^{5} : 2^{3} = 2^{2} = 4$
Rule: Power of power → Formula: $(a^{m})^{n} = a^{m \cdot n}$ → Example: $(2^{3})^{2} = 2^{6} = 64$
Rule: Negative exponent → Formula: $a^{- n} = \frac{1}{a ^{n}}$ → Example: $2^{- 3} = \frac{1}{8}$
Rule: Zero exponent → Formula: $a^{0} = 1$ → Example: $5^{0} = 1$
Rule: Product to power → Formula: $(a \cdot b)^{n} = a^{n} \cdot b^{n}$ → Example: $(2 \cdot 3)^{2} = 36$

Powers Examples: Solved Problems and Exercises

Table of Contents

Basic examples

Calculate the power

Examples with rules of powers

Multiplying powers with the same base

Dividing powers with the same base

Raising a power to a power

Examples with negative exponents

Converting to fraction

Mixed operations

Examples with decimal numbers

Squaring decimals

Decimal exponents

Examples with negative base

Even exponent

Odd exponent

Comparison

Equations with powers

Simple equation: $x^{2} = 16$

Equation with exponents: $2^{x} = 32$

Equation with negative exponent: $3^{x} = \frac{1}{9}$

Practical problems

Problem 1: Area of a square

Problem 2: Distance and speed

Problem 3: Population growth

Interactive exercises

Summary of formulas

Powers Examples: Solved Problems and Exercises

Table of Contents

Basic examples

Calculate the power

Examples with rules of powers

Multiplying powers with the same base

Dividing powers with the same base

Raising a power to a power

Examples with negative exponents

Converting to fraction

Mixed operations

Examples with decimal numbers

Squaring decimals

Decimal exponents

Examples with negative base

Even exponent

Odd exponent

Comparison

Equations with powers

Simple equation: x2=16

Equation with exponents: 2x=32

Equation with negative exponent: 3x=91​

Practical problems

Problem 1: Area of a square

Problem 2: Distance and speed

Problem 3: Population growth

Interactive exercises

Summary of formulas

Simple equation: $x^{2} = 16$

Equation with exponents: $2^{x} = 32$

Equation with negative exponent: $3^{x} = \frac{1}{9}$