how to solve problems with exponents

How to solve exponential equations

Examples of exponential equations.

$$ 2^{\red x} = 4 \\ 8^{\red{2x}} = 16 \\ \\ 16^{\red { x+1}} = 256 \\ \left( \frac{1}{2} \right)^{\red { x+1}} = 512 $$

As you might've noticed, an exponential equation is just a special type of equation. It's an equation that has exponents that are $$ \red{ variables}$$.

Steps to Solve

There are different kinds of exponential equations. We will focus on exponential equations that have a single term on both sides. These equations can be classified into 2 types.

$$ 4^x = 4^9 $$.

$$ 4^3 = 2^x $$.

$$ \left( \frac{1}{4} \right)^x = 32 $$ (Part II below)

Part I. Solving Exponential Equations with Same Base

Solve: $$ 4^{x+1} = 4^9 $$

Ignore the bases, and simply set the exponents equal to each other

$$ x + 1 = 9 $$

Solve for the variable

$$ x = 9 - 1 \\ x = \fbox { 8 } $$

$$ 4^{x+1} = 4^9 \\ 4^{\red{8}+1} = 4^9 $$ $$ 4^{\red{9} } = 4^9 $$

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II. Solving Exponential Equations with un -like bases

What do they look like.

$$ \red 4^3 = \red 2^x $$ $$ \red 9^x = \red { 81 } $$ $$ \left( \red{\frac{1}{2}} \right)^{ x+1} = \red 4^3 $$ $$ \red 4^{2x} +1 = \red { 65 } $$

In each of these equations, the base is different. Our goal will be to rewrite both sides of the equation so that the base is the same.

Solve: $$ 4^{3} = 2^x $$

Answer : They are both powers of 2

Rewrite equation so that both exponential expressions use the same base

$$ \red 4^{3} = 2^x \\ (\red {2^2})^{3} = 2^x $$

Use exponents laws to simplify

$$ (\red {2^2})^{3} = 2^x \\ (2^\red {2 \cdot 3 }) = 2^x \\ (2^\red 6 ) = 2^x $$

Solve like an exponential equation of like bases

$$ (2^\red 6 ) = 2^x \\ x = \fbox{6} $$

Substitute $$\red 6 $$ into the original equation to verify our work.

$$ 4^{3} = 2^{\red 6} $$ $$ 64 = 64 $$

Example with Negative Exponent

Unlike bases often involve negative or fractional bases like the example below. We are going to treat these problems like any other exponential equation with different bases--by converting the bases to be the same.

Practice Problems ( un -like bases)

Solve the following exponential Equation: $$9^x = 81$$

You can use either 3 or 9 . I will use 9.

$ \\ 81 = \red 9 ^{\blue 2} \\ 9 = \red 9 ^{\blue 1} \\ $

Substitute the rewritten bases into original equation

$$ (\red 9^{\blue 1})^x = \red 9^{\blue 2} $$ 2 ) x =3 4 -->

$$ (\red 9^{\blue 1})^x = \red 9^{\blue 2} \\ 9^{1 \cdot x } = 9 ^{2} \\ 9^{x } = 9 ^{2} $$ 2 x =3 4 -->

$$ x = 2 $$ -->

Solve the equation : $$ 4^{2x} +1 = 65 $$

Rewrite this equation so that it looks like the other ones we solved. Isolate the exponential expression as follows:

$$ 4^{2x} +1 \red{-1} = 65\red{-1} \\ 4^{2x} = 64 $$

They are both powers of 2 and of 4 . You could use either base to solve this. I will use base 4

$ \\ 64 = \red 4 ^{\blue 3} \\ 4 = \red 4 ^{\blue 1} \\ $

$$ 4^{2x} = 64 \\ \red 4^{\blue{ 2x }} = \red 4^{\blue 3 } $$

$$ \red 4^{\blue{ 2x }} = \red 4^{\blue 3 } $$ 2x = 4 3 -->

Not much to do this time :)

$$ 2x = 3 \\ x = \frac{3}{2} $$

Solve the exponential Equation : $$ \left( \frac{1}{4} \right)^x = 32 $$

Since these equations have different bases, follow the steps for unlike bases

They are both powers of 2

$ \\ 32 = \red 2 ^{\blue 5} \\ \frac 1 4 = \red 2 ^{\blue {-2}} \\ $

Rewrite as a negative exponent and substitute the rewritten bases into original equation

$$ \left( \frac{1}{4} \right)^x = 32 \\ \left( \frac{1}{2^2} \right)^x = 32 \\ \left(\red 2 ^{\blue{-2}} \right)^x = \red 2^{\blue 5} $$

$$ \left(\red 2 ^{\blue{-2}} \right)^x = \red 2^{\blue 5} \\ 2 ^{-2 \cdot x} = 2^5 \\ 2 ^{-2x} = 2^5 $$

$$ 2 ^{-2x} = 2^5 \\ -2x = 5 \\ \frac{-2x}{-2} = \frac{5}{-2} \\ x = -\frac{5}{2} $$

Solve this exponential equation: $$ \left( \frac{1}{9} \right)^x-3 = 24 $$

$$ \left( \frac{1}{9} \right)^x -3 \red{+3} =24\red{+3} \\ \left( \frac{1}{9} \right)^x=27 $$

They are both powers of 3 .

$ \\ \frac 1 9 = \red 3 ^{\blue {-2}} \\ 27 = \red 3 ^{\blue 3} \\ $

Rewrite as a negative exponent and substitute into original equation

$$ \left( \frac{1}{9} \right)^x=27 \\ \left( \red{3^{-2}}\right)^x=\red{3^3 } $$

$$ 3^\red{{-2 \cdot x}} = 3^3 \\ 3^\red{{-2x}} = 3^3 $$

$$ -2x = 3 \\ x = \frac{3}{-2} \\ x = -\frac{3}{2} $$

Solve this exponential equation: $$ \left( \frac{1}{25} \right)^{(3x -4)} -1 = 124 $$

Rewrite this equation so that it looks like the other ones we solved--In other words, isolate the exponential expression as follows:

$$ \left( \frac{1}{25} \right)^{(3x -4)} -1 \red{+1} = 124 \red{+1} \\ \left( \frac{1}{25} \right)^{(3x -4)} = 125 $$

They are both powers of 5 .

$ \\ \frac { 1 } { 25 } = \red 5 ^{\blue {-2}} \\ 125 = \red 5 ^{\blue 3} \\ $

$$ \left( \frac{1}{25} \right)^{(3x -4)} = 125 \\ \left( \red{5^{-2}} \right)^{(3x -4)} = \red{5^3} $$

$$ 5^\red{{-2 \cdot (3x -4)}} = 5^3 \\ 5^\red{{(-6x + 8)}} = 5^3 $$

$$ -6x + 8 =3 \\ -6x = -5 \\ x = \frac{-5}{-6} \\ x = \frac{5}{6} $$

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How to Solve Exponents: A Step-by-Step Guide with Examples

Exponents, also known as powers or indices, are a fundamental concept in mathematics. It plays a crucial role in various fields, from basic arithmetic to advanced calculus. An exponent represents how many times a number, called the base, is multiplied by itself. To solve exponents involves understanding the properties of exponents, simplifying expressions, and performing operations with exponential terms. In this guide, we will walk you through the process of solving exponent problems using maths.ai as a teaching tool.

Understanding Exponents

Before we delve into solving exponents problems, let’s review the basics of exponents.

An exponent is usually written as a superscript number next to the base. For example, in the expression 2^3, 2 is the base, and 3 is the exponent. This means that 2 is multiplied by itself three times: 2 * 2 * 2 = 8.

Example 1: Simple Exponentiation

Let’s solve the problem: 4^2.

Step 1: Identify the base and the exponent.

Exponent: 2

Step 2: Apply the exponentiation rule.

4^2 = 4 * 4 = 16

Exponent Properties

Understanding exponent properties is essential for solving more complex exponent problems.

Property 1: Product of Powers

a^m * a^n = a^(m + n)

This property states that when you multiply two powers with the same base, you can add their exponents.

Example 2: Product of Powers

Solve the problem: 3^4 * 3^2.

Step 1: Identify the base and the exponents.

Exponent 1: 4

Exponent 2: 2

Step 2: Apply the product of powers property.

3^4 * 3^2 = 3^(4 + 2) = 3^6 = 729

Property 2: Quotient of Powers

a^m / a^n = a^(m – n)

This property states that when you divide two powers with the same base, you can subtract their exponents.

Example 3: Quotient of Powers

Solve the problem: 5^7 / 5^4.

Exponent 1: 7

Exponent 2: 4

Step 2: Apply the quotient of powers property.

5^7 / 5^4 = 5^(7 – 4) = 5^3 = 125

Property 3: Power of a Power

(a^m)^n = a^(m * n)

This property states that when you raise a power to another power, you can multiply the exponents.

Example 4: Power of a Power

Solve the problem: (2^3)^4.

Exponent: 4

Step 2: Apply the power of a power property.

(2^3)^4 = 2^(3 * 4) = 2^12 = 4096

Simplifying Exponential Expressions

Simplifying exponential expressions involves applying the properties of exponents to combine or rewrite terms for easier computation.

Example 5: Simplifying Exponential Expressions

Simplify the expression: 2^5 * 2^8 / 2^3.

Exponent 1: 5, Exponent 2: 8, Exponent 3: 3

Step 2: Apply the properties of exponents.

2^5 * 2^8 / 2^3 = 2^(5 + 8 – 3) = 2^10 = 1024

Operations with Exponential Terms

Performing operations with exponential terms involves applying the properties of exponents while considering addition, subtraction, multiplication, and division.

Example 6: Operations with Exponential Terms

Solve the expression: (4^3 * 4^2) / 4^4 + 4^1.

Step 1: Identify the base and the exponents for each term.

Term 1: 4^3 * 4^2,Term 2: 4^4, Term 3: 4^1

Step 2: Simplify each term using the properties of exponents.

Term 1: 4^3 * 4^2 = 4^(3 + 2) = 4^5, Term 2: 4^4, Term 3: 4^1

Step 3: Combine the terms using addition and subtraction.

(4^5 * 4^4) / 4^4 + 4^1 = 4^5 + 4^1

Step 4: Calculate the final result.

4^5 + 4^1 = 1024 + 4 = 1028

Exponents are a fundamental concept in mathematics that allows us to represent repeated multiplication in a concise form. By understanding the properties of exponents, simplifying expressions, and performing operations with exponential terms, you can confidently tackle various exponent problems. Remember to apply the rules systematically, and when in doubt, use tools maths.ai to assist you in your journey towards mastering exponents. With practice and a clear grasp of exponent principles, you’ll be well-equipped to solve more complex maths problems that involve exponents.

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How to Solve Exponential Equations

Last Updated: November 25, 2022 Fact Checked

This article was co-authored by David Jia . David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. There are 8 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 133,413 times.

Exponential equations may look intimidating, but solving them requires only basic algebra skills. Equations with exponents that have the same base can be solved quickly. In other instances, it is necessary to use logs to solve. Even this method, however, is simple with the aid of a scientific calculator.

Equating Two Exponents with the Same Base

Equating an Exponent and a Whole Number

Using Logs for Terms without the Same Base

Expert Q&A

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• ↑ https://www.mathsisfun.com/definitions/exponent.html
• ↑ http://www.mathwarehouse.com/algebra/exponents/solve-exponential-equations-how-to.php
• ↑ https://courses.lumenlearning.com/intermediatealgebra/chapter/use-compound-interest-formulas/
• ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut45_expeq.htm
• ↑ http://www.purplemath.com/modules/solvexpo.htm
• ↑ http://www.mathsisfun.com/algebra/exponents-logarithms.html
• ↑ https://tutorial.math.lamar.edu/classes/alg/solveexpeqns.aspx
• ↑ https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:exp-eq-log/a/solving-exponential-equations-with-logarithms

About This Article

To solve exponential equations with the same base, which is the big number in an exponential expression, start by rewriting the equation without the bases so you're left with just the exponents. Then, solve the new equation by isolating the variable on one side. To check your work, plug your answer into the original equation, and solve the equation to see if the two sides are equal. If they are, your answer is correct. To learn how to solve exponential equations with different bases, scroll down! Did this summary help you? Yes No

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The exponent of a number says how many times to use the number in a multiplication.

In 8 2 the "2" says to use 8 twice in a multiplication, so 8 2 = 8 × 8 = 64

In words: 8 2 could be called "8 to the power 2" or "8 to the second power", or simply "8 squared"

Example: 5 3 = 5 × 5 × 5 = 125

• In words: 5 3 could be called "5 to the third power", "5 to the power 3" or simply "5 cubed"

Example: 2 4 = 2 × 2 × 2 × 2 = 16

• In words: 2 4 could be called "2 to the fourth power" or "2 to the power 4" or simply "2 to the 4th"

Exponents make it easier to write and use many multiplications

Example: 9 6 is easier to write and read than 9 × 9 × 9 × 9 × 9 × 9

You can multiply any number by itself as many times as you want using exponents.

So in general :

Another Way of Writing It

Sometimes people use the ^ symbol (above the 6 on your keyboard), as it is easy to type.

Example: 2^4 is the same as 2 4

• 2^4 = 2 × 2 × 2 × 2 = 16

Negative Exponents

Negative? What could be the opposite of multiplying? Dividing!

So we divide by the number each time, which is the same as multiplying by 1 number

Example: 8 -1 = 1 8 = 0.125

We can continue on like this:

Example: 5 -3 = 1 5 × 1 5 × 1 5 = 0.008

But it is often easier to do it this way:

5 -3 could also be calculated like:

1 5 × 5 × 5 = 1 5 3 = 1 125 = 0.008

Negative? Flip the Positive!

More Examples:

What if the Exponent is 1, or 0?

It all makes sense.

If you look at that table, you will see that positive, zero or negative exponents are really part of the same (fairly simple) pattern:

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Exponents And Powers

Exponents and powers are ways used to represent very large numbers or very small numbers in a simplified manner. For example, if we have to show 3 x 3 x 3 x 3 in a simple way, then we can write it as 3 4 , where 4 is the exponent and 3 is the base. The whole expression 3 4 is said to be power. Also learn  the  laws of exponents here.

Basically, power is an expression that shows repeated multiplication of the same number or factor. The value of the exponent is based on the number of times the base is multiplied to itself. See of the examples here:

What is Exponent?

An exponent of a number, represents the number of times the number is multiplied to itself. If 8 is multiplied by itself for n times, then, it is represented as:

8 x 8 x 8 x 8 x …..n times = 8 n

The above expression, 8 n , is said as 8 raised to the power n. Therefore, exponents are also called power or sometimes indices.

• 2 x 2 x 2 x 2 = 2 4
• 5 x 5 x 5 = 5 3
• 10 x 10 x 10 x 10 x 10 x 10 = 10 6

General Form of Exponents

The exponent is a simple but powerful tool. It tells us how many times a number should be multiplied by itself to get the desired result. Thus any number ‘a’ raised to power ‘n’ can be expressed as:

Here a is any number and n is a natural number.

a n  is also called the nth power of a .

‘a’ is the base and ‘n’ is the exponent or index or power.

 ‘a’ is multiplied ‘n’ times, and thereby exponentiation is the shorthand method of repeated multiplication.

Also, read:

• Exponents and Powers for Class 7
• Exponents And Powers for Class 8

Laws of Exponents

The laws of exponents are demonstrated based on the powers they carry.

• Bases – multiplying the like ones – add the exponents and keep the base same. (Multiplication Law)
• Bases – raise it with power to another – multiply the exponents and keep the base same.
• Bases – dividing the like ones – ‘Numerator Exponent – Denominator Exponent’ and keep the base same. (Division Law)

Let ‘a’ is any number or integer (positive or negative) and ‘m’,  ‘n’ are positive integers, denoting the power to the bases, then;

Multiplication Law

As per the multiplication law of exponents, the product of two exponents with the same base and different powers equals to base raised to the sum of the two powers or integers.

           a m  × a n   = a m+n

Division Law

When two exponents having same bases and different powers are divided, then it results in base raised to the difference between the two powers.

          a m ÷ a n   = a m  / a n   = a m-n

Negative Exponent Law

Any base if has a negative power, then it results in reciprocal but with positive power or integer to the base.

            a -m    = 1/a m  

Rules of Exponents

The rules of exponents are followed by the laws. Let us have a look at them with a brief explanation.

Suppose ‘a’ & ‘b’ are the integers and ‘m’ & ‘n’ are the values for powers, then the rules for exponents and powers are given by:

i) a 0  = 1

As per this rule, if the power of any integer is zero, then the resulted output will be unity or one.

Example: 5 0 = 1

ii) (a m ) n  = a( mn )

‘a’ raised to the power ‘m’ raised to the power ‘n’ is equal to ‘a’ raised to the power product of ‘m’ and ‘n’.

Example: (5 2 ) 3  = 5 2 x 3

iii) a m  × b m  =(ab) m

The product of ‘a’ raised to the power of ‘m’ and ‘b’ raised to the power ‘m’ is equal to the product of ‘a’ and ‘b’ whole raised to the power ‘m’.

Example:  5 2  × 6 2  =(5 x 6) 2

iv) a m /b m  = (a/b) m

The division of ‘a’ raised to the power ‘m’ and ‘b’ raised to the power ‘m’ is equal to the division of ‘a’ by ‘b’ whole raised to the power ‘m’.

Example:  5 2 /6 2  = (5/6) 2

Solved Questions

Example 1: Write 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 in exponent form.

In this problem 7s are written 8 times, so the problem can be rewritten as an exponent of 8.

7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 = 7 8 .

Example 2 : Write below problems like exponents:

• 3 x 3 x 3 x 3 x 3 x 3
• 7 x 7 x 7 x 7 x 7
• 10 x 10 x 10 x 10 x 10 x 10 x 10
• 3 x 3 x 3 x 3 x 3 x 3 = 3 6
• 7 x 7 x 7 x 7 x 7 = 7 5
• 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10 7  

Example 3: Simplify 25 3 /5 3  

 Using Law: a m /b m  = (a/b) m

25 3 /5 3   can be written as (25/5) 3   = 5 3  = 125.

Exponents and Powers Applications

Scientific notation uses the power of ten expressed as exponents, so we need a little background before we can jump in. In this concept, we round out your knowledge of exponents, which we studied in previous classes.

The distance between the Sun and the Earth is 149,600,000 kilometres. The mass of the Sun is 1,989,000,000,000,000,000,000,000,000,000 kilograms. The age of the Earth is 4,550,000,000 years. These numbers are way too large or small to memorize in this way.  With the help of exponents and powers, these huge numbers can be reduced to a very compact form and can be easily expressed in powers of 10.

Now, coming back to the examples we mentioned above, we can express the distance between the Sun and the Earth with the help of exponents and powers as following:

Distance between the Sun and the Earth 149,600,000 = 1.496× 10 × 10 × 10 × 10 × 10× 10 × 10 = 1.496× 10 8 kilometers.

Mass of the Sun: 1,989,000,000,000,000,000,000,000,000,000 kilograms = 1.989 × 10 30  kilograms.

Age of the Earth:  4,550,000,000 years = 4. 55× 10 9  years

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Course: Algebra 2   >   Unit 6

Evaluating fractional exponents.

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