Powers and Exponents

Q: What are the laws of exponents?

The laws of exponents are as follows: One as an exponent: Any number or variable raised to the first power equals the same number or variable. Zero as an exponent: Any number or variable raised to zero power equals 1. Negative exponent: Any number or variable raised to a negative exponent equals its reciprocal, which is 1 over the same number or variable with a positive exponent. Power of a power: If you have a power raised to another power, then you can simplify this expression by multiplying the exponents. Product: a) Same base: If you have the product of two powers with the same base, then you can combine them by adding the exponents together. b) Different base: If you have the product of two powers with different bases and exponents, then you need to solve each power separately, then multiply the results together. Quotient: If you have the quotient of two powers with the same base, then you can combine them by subtracting the exponent in the numerator minus the exponent in the denominator. Power of a product: If you have the power of a product, then you can distribute the exponent to each factor. Power of a quotient: If you have the power of a quotient, then you can distribute the exponent to the numerator and the denominator.

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Powers and exponents are terms that can cause confusion, as sometimes they are used interchangeably. However, in this article, we will explain their official definition and the meaning behind them, as well as the different laws that you can use to solve problems involving powers in Algebra using practical examples.

Meaning of Powers and Exponents

Powers are mathematical expressions in the form $x^{n}$ , where x is the base and n is the exponent. Powers represent repeated multiplication, where the base is the number or variable that is multiplied repeatedly, and the exponent indicates how many times to multiply the base by itself.

From the above definition, let's identify each part of a power and its meaning:

Powers and Exponents Power representation StudySmarter Parts of a power, StudySmarter Originals

The expression $x^{n}$ can be read as x to the $n^{t h}$ power. The base (x) is written in full size, and the exponent (n) is written smaller and at the top of the line using superscript (if you are typing it on a computer). For example, $x^{2}$ is read as x to the second power, or x squared, which in practice means that the value of x is multiplied by itself as many times as the value of the exponent, which in this case is 2, like this:

$x^{2} = x \cdot x$ .

Remember: The value of the exponent tells you how many times the base will be repeated in the multiplication. If a number or variable has no exponent, then it is assumed to be 1. For example, $x^{1} = x$ . Also, any number or variable with an exponent of 0 (zero) is equal to 1. For example, $x^{0} = 1$ .

Examples of powers

Here are some examples of powers, including how you can read the expressions and their meaning:

Power	In words	Meaning
$3^{1}$	Three to the first power	$3$
$5^{2}$	Five to the second power, or five squared	$5 \cdot 5$
$2^{3}$	Two to the third power, or two cubed	$2 \cdot 2 \cdot 2$
$7^{4}$	Seven to the fourth power	$7 \cdot 7 \cdot 7 \cdot 7$
$10^{5}$	Ten to the fifth power	$10 \cdot 10 \cdot 10 \cdot 10 \cdot 10$
$x^{n}$	x to the n^th power	$x \cdot x \cdot x \cdot . . . \cdot x$ x is repeated as many times as the value of n

How to Solve Powers

To solve powers, you need to consider two different situations:

1. If the base is a number: In this case, all you need to do is multiply the base by itself as many times as the value of the exponent to find the solution.

Solve $3^{3}$

3^{3} = 3 \cdot 3 \cdot 3 = 27

2. If the base is a variable: In this case, you need to substitute the variable with a value and then proceed as before.

Solve $x^{5}$ , when x = 2

$x^{5} = 2^{5} = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$

Laws of Exponents

The different laws of exponents that you can use to solve problems in Algebra are as follows:

Name	Law	Description	Example
One as an exponent	$x^{1} = x$	Any number or variable raised to the first power equals the same number or variable.	$2^{1} = 2$
Zero as an exponent	$x^{0} = 1$	Any number or variable raised to zero power equals 1.	$5^{0} = 1$
Negative exponent	$x^{- n} = \frac{1}{x^{n}}$	Any number or variable raised to a negative exponent equals its reciprocal, which is 1 over the same number or variable with a positive exponent.	$x^{- 2} = \frac{1}{x^{2}}$
Power of a Power	${(x^{m})}^{n} = x^{m \cdot n}$	If you have a power raised to another power, you can simplify this expression by multiplying the exponents.	${(x^{2})}^{3} = x^{2 \cdot 3} = x^{6}$
Product	Same base $x^{m} \cdot x^{n} = x^{m + n}$	If you have the product of two powers with the same base, then you can combine them by adding the exponents together.	$x^{3} \cdot x^{5} = x^{3 + 5} = x^{8}$
Product	Different base $x^{m} \cdot y^{n} = (x^{m}) \cdot (y^{n})$	If you have the product of two powers with different bases and exponents, then you need to solve each power separately, and then multiply the results together.	$2^{3} \cdot 3^{2} = (2^{3}) \cdot (3^{2}) = (2 \cdot 2 \cdot 2) \cdot (3 \cdot 3) = 8 \cdot 9 = 72$
Quotient	$\frac{x^{m}}{x^{n}} = x^{m - n}$	If you have the quotient of two powers with the same base, you can combine them by subtracting the exponent in the numerator minus the exponent in the denominator.	$\frac{x^{6}}{x^{2}} = x^{6 - 2} = x^{4}$
Power of a Product	${(x \cdot y)}^{n} = x^{n} \cdot y^{n}$	If you have the power of a product, then you can distribute the exponent to each factor.	${(x \cdot y)}^{5} = x^{5} \cdot y^{5}$
Power of a quotient	${(\frac{x}{y})}^{n} = \frac{x^{n}}{y^{n}}$	If you have the power of a quotient, then you can distribute the exponent to the numerator and the denominator.	${(\frac{x}{y})}^{4} = \frac{x^{4}}{y^{4}}$

Solving Powers and Exponents

Sometimes you will need to use more than one law of exponents to be able to solve more complex problems:

1. Evaluate or simplify $\frac{24 x^{4} y^{5}}{4 x^{5}}$

$\frac{24 x^{4} y^{5}}{4 x^{5}} = 6 x^{- 1} y^{5}$ using the quotient law $\frac{x^{m}}{x^{n}} = x^{m - n}$

$\frac{24 x^{4} y^{5}}{4 x^{5}} = \frac{6 y^{5}}{x}$ using the negative exponent law $x^{- n} = \frac{1}{x^{n}}$

2. Evaluate or simplify ${(\frac{3 x y^{2}}{2 x^{3}})}^{- 2}$

${(\frac{3 x y^{2}}{2 x^{3}})}^{- 2} = {(\frac{2 x^{3}}{3 x y^{2}})}^{2}$ using the negative exponent law $x^{- n} = \frac{1}{x^{n}}$ , flip the fraction

$= \frac{{(2 x^{3})}^{2}}{{(3 x y^{2})}^{2}}$ using the power of a quotient law ${(\frac{x}{y})}^{n} = \frac{x^{n}}{y^{n}}$ , distribute the exponent

$= \frac{4 x^{6}}{9 x^{2} y^{4}}$ using the quotient law $\frac{x^{m}}{x^{n}} = x^{m - n}$

${(\frac{3 x y^{2}}{2 x^{3}})}^{- 2} = \frac{4 x^{4}}{9 y^{4}}$

Application of Powers and Exponents

You must be thinking, powers and exponents are very interesting, but where and when would I need to used them? Let's give you an idea of the diverse applications of powers and exponents in real-life.

Powers and exponents are used in many areas, for example:

We can use powers and exponents to represent very small numbers and very large numbers too, with a simpler mathematical notation.
The most evident application of powers and exponents is when we calculate measurements like area or volume, as you might recall, when you calculate area, the result is given in square units, such as cm² or m², and volume is given in cubic units like cm³ or m³.
Scientific scales, like pH scale and Richter scale, also use powers and exponents to represent the different levels on their scales. As going up or down on the scales represent an increase or reduction of 10 times the previous level.
Computer games physics is another application of powers and exponents, helping to calculate movement, interaction between objects and other games dynamics.
The capacity of computer memory is also represented using powers and exponents.
Architecture uses powers and exponents to specify the scale of the models created by an architect in comparison to the size of the structure or building in real-life.
Other fields where powers and exponents are used include economics, accounting and finance, among others.

Powers and Exponents - key takeaways

Powers are mathematical expressions in the form $x^{n}$ , where x is known as the base and n is the exponent.
Powers represent repeated multiplication, where the base is the number or variable that is multiplied repeatedly, and the exponent indicates how many times the base will be repeated in the multiplication.
To solve powers, consider if the base is a number or a variable. If the base is a number, multiply the base by itself as many times as the value of the exponent to find the solution. If the base is a variable, substitute the variable with a value and proceed as before.
The different laws of exponents can be used to help solve problems in algebra.

Frequently Asked Questions about Powers and Exponents

Powers are mathematical expressions in the form xⁿ, where x is known as the base and n is the exponent. Powers represent repeated multiplication, where the base is the number or variable that is multiplied repeatedly, and the exponent indicates how many times to multiply the base by itself.

To solve powers, consider if the base is a number or a variable. If the base is a number, multiply the base by itself as many times as the value of the exponent to find the solution. If the base is a variable, substitute the variable with a value, and then proceed as before.

Examples of powers include: 2³ and 5² .

The laws of exponents are as follows:

One as an exponent: Any number or variable raised to the first power equals the same number or variable.
Zero as an exponent: Any number or variable raised to zero power equals 1.
Negative exponent: Any number or variable raised to a negative exponent equals its reciprocal, which is 1 over the same number or variable with a positive exponent.
Power of a power: If you have a power raised to another power, then you can simplify this expression by multiplying the exponents.
Product:

a) Same base: If you have the product of two powers with the same base, then you can combine them by adding the exponents together.

b) Different base: If you have the product of two powers with different bases and exponents, then you need to solve each power separately, then multiply the results together.

Quotient: If you have the quotient of two powers with the same base, then you can combine them by subtracting the exponent in the numerator minus the exponent in the denominator.
Power of a product: If you have the power of a product, then you can distribute the exponent to each factor.
Power of a quotient: If you have the power of a quotient, then you can distribute the exponent to the numerator and the denominator.

How to multiply exponents with different bases and powers: if you have the product of two powers with different bases and exponents, then you need to solve each power separately, then multiply the results together.

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What is scientific notation?

Scientific notation is a a way to rewrite multi-digit numbers in a compact way in the form $a\times 10^b$ where $1<|a|<10$ and $b$ is an integer.

A number can be said to be written in scientific notation when a number between 1 and 10 is multiplied by a power of...

10.

The format for scientific notation is written as $a\times 10^b$.

True.

The exponent of $10$ in scientific notation is an integer.

False.

What is the value range a coefficient for scientific notation could exist ?

$1\leq a\leq 10$, where a is the coefficient.

Write $0.00097$ in scientific notation.

$9.7\times 10^{-4}$.

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Powers and Exponents

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Powers and Exponents

Meaning of Powers and Exponents

Examples of powers

How to Solve Powers

Laws of Exponents

Solving Powers and Exponents

Application of Powers and Exponents

Powers and Exponents - key takeaways

Frequently Asked Questions about Powers and Exponents

What are powers and exponents?

How do you solve exponents and powers?

What are examples of powers and exponents?

What are the laws of exponents?

How do you multiply exponents with different bases and powers?

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